A Tighter Insertion-based Approximation of the Crossing Number

A Tigh ter Insertion-based Appro ximation of the Crossing Num b er ? Markus Chimani 1 and P etr Hlinˇ en´ y ?? 2 1 Theoretical Computer Science, Universit y Osnabr¨ uck, German y markus.chimani@uni-osnabrueck.de 2 F acult y of Informatics, Masaryk Universit y Brno, Czech Republic hlineny@fi.muni.cz Septem b er 13, 2021 Abstract. Let G b e a planar graph and F a set of additional edges not y et in G . The multiple e dge insertion problem (MEI) asks for a drawing of G + F with the minimum n umber of pairwise edge crossings, such that the sub dra wing of G is plane. Finding an exact solution to MEI is NP-hard for general F . W e present the ﬁrst p olynomial time algorithm for MEI that achiev es an additive approximation guarantee – dep ending only on the size of F and the maximum degree of G , in the case of connected G . Our algorithm seems to b e the ﬁrst directly implementable one in that realm, to o, next to the single edge insertion. It is also kno wn that an (ev en appro ximate) solution to the MEI problem w ould approximate the crossing num b er of the F -almost-planar gr aph G + F , while computing the crossing n umber of G + F exactly is NP-hard already when | F | = 1. Hence our algorithm induces new, improv ed ap- pro ximation b ounds for the crossing n umber problem of F -almost-planar graphs, achieving constant-factor appro ximation for the large class of suc h graphs of b ounded degrees and b ounded size of F . 1 In tro duction The crossing n um b er cr( G ) of a graph G is the minimum num ber of pairwise edge crossings in a drawing of G in the plane. Finding the cossing n umber of a graph is one of the most prominen t combinatorial optimization problems in graph theory and is NP-hard already in very restricted cases, e.g., even when considering a planar graph with one added edge [4]. It has b een vividly in vestigated for ov er 60 y ears, but there is still surprisingly little kno wn ab out it; see [25] for an extensiv e bibliograph y . While the appro ximability status of the crossing n umber problem is still unknown, several approximation algorithms arose for sp ecial graph classes. F or the crossing num b er of general graphs with bounded degree, there is an algorithm [1] that appro ximates not directly the crossing num b er, but the quan tity n + cr( G ); its currently b est incarnation do es so within a factor of ? A preliminary version of this research app eared in ICALP 2011 [8]. ?? P . Hlinˇ en´ y has b een supp orted by the Czec h Science F oundation pro ject 14-03501S. O (log 2 n ) [13]. In terms of pure cr( G )-appro ximation this hence resembles a ratio of O ( n log 2 n ). The ﬁrst sublinear approximation factor of ˜ O ( n 0 . 9 ) was recen tly given via a highly inv olved (and unfortunately not directly practical) algorithm [10]. The known constant factor appro ximations restrict themselv es to graphs fol- lo wing one of tw o paradigms (see also Section 4): they either assume that the graph is embeddable in some higher surface [14, 19, 21], or they are based on the idea that only a small set of graph elements has to b e remo ved from G to mak e it planar: remo ving and re-inserting them can give strong approximation b ounds [3, 9, 20]. In this pap er, we follow the latter idea and ﬁrst concentrate on the follo wing tightly related problem: Deﬁnition 1.1 (Multi ple edge insertion, MEI). Let G b e a planar graph and F a set of edges (vertex pairs, in fact) not in E ( G ). W e denote by G + F the graph obtained by adding F to the edge set of G . The multiple e dge insertion problem MEI( G, F ) is to ﬁnd i) a plane em b edding G 0 of G , and ii) a dra wing G F of the graph G + F such that the restriction of G F to G is the plane em b edding G 0 , suc h that the n umber of pairwise edge crossings in G F is minimized ov er all plane em b eddings G 0 of G and all dra wings G F as in (ii). Let ins( G 0 , F ) denote the minimum n umber of edge crossings of G F as in (ii) and let ins( G, F ), the solution size of the MEI( G, F ) problem, denote this min- im um of ins( G 0 , F ) ov er all plane embeddings G 0 of G .  W e refer to Section 2 for formal deﬁnitions of a drawing and edge crossings. Observ e that, in a solution to MEI, each crossing hence in v olves at least one edge of F . This c an b e a severe restriction, as shown b y the examples of MEI( G, F ) instances with solution sizes m uch larger than cr( G + F ) given, e.g., in [18, 20]. F or general k = | F | , the MEI problem is known to be NP-hard [26], based on a reduction from ﬁxe d line ar cr ossing numb er (see App endix); for ﬁxed k > 1 the problem complexity is op en. The main diﬃculty of the MEI problem, roughly , comes from p ossible existence of (up to exp onen tially many) inequiv alen t em- b eddings of G . The case k = 1 of MEI is kno wn as the (single) e dge insertion problem and can b e solv ed optimally in linear time [18], as w e will brieﬂy summarize in Section 2.5. Let e b e the edge to insert, and denote the resulting num b er of crossings by ins( G, e ). Let ∆ ( G ) denote the maximum degree in G . It was sho wn [3, 20] that ins( G, e ) approximates the crossing num b er cr( G + e )—i.e., of the graph con taining this edge e —within a multiplicativ e factor of b 1 2 ∆ ( G ) c ac hieved in [3], and this b ound is tight. Recall also that computing cr( G + e ) exactly is NP-hard [4]. Another sp ecial case of the MEI problem is when one adds a new vertex together with its incident edges; this is also p olynomially solv able [7] and ap- pro ximates the crossing num b er of the resulting ap ex gr aph [9]. A slight v ariant 2 of this vertex insertion (or star insertion ) problem, is the multiple adjac ent e dge insertion , were the inserted edges hav e a common inciden t vertex that need not b e new. The algorithm in [7] also solves this latter v ariant. How ever, these are the only t yp es of insertion problems that are currently kno wn to b e in P . Nev ertheless, it has b een prov en in [9] (see Section 4) that a solution (even an approximate one) to MEI( G, F ) w ould directly imply an approximation al- gorithm for cr( G + F ) w ith planar G . Indep enden tly , Chuzho y et al. [11] hav e sho wn the ﬁrst algorithm eﬃciently computing an approximate solution to the crossing num b er problem on G + F with the help of a m ultiple edge insertion solution. Precisely , they hav e achiev ed a solution with the num b er of crossings cr aprx ( G + F ) ≤ O  ∆ ( G ) 3 · | F | · cr( G + F ) + ∆ ( G ) 3 · | F | 2  (1) (without giving explicit constan ts). Though not mentioned explicitly in [11], it seems that their results also give an appro ximation solution to MEI( G, F ) with the same ratio, at least in the case of 3-connected G + F . How ev er, the algorithm [11] is unfortunately not directly applicable in practice. In this pap er, w e pay our main attention to the MEI problem. In contrast to a multipl icative factor as in (1), we pro vide an eﬃcien t algorithm approximating a solution of MEI( G, F ) with an additive approximation guarantee (3). Then w e emplo y the aforemen tioned generic result of [9] to derive a corresponding appro ximation of the crossing num b er (this up to a multiplicativ e factor). On the one hand, our approach is algorithmically and implementationally simpler, virtually only building on top of w ell-studied and exp erimen tally ev aluated sub- algorithms. On the other hand, it gives stronger approximations also for the crossing n umber, cf. (5) as compared to (1), as well as b etter run time b ounds. W e are going to show: Theorem 1.2. Given a c onne cte d planar gr aph G and an e dge set F , F ∩ E ( G ) = ∅ . L et k := | F | and ∆ := ∆ ( G ) . Algorithm 3.7 describ e d b elow ﬁnds, in O ( k · | V ( G ) | + k 2 ) time, a solution to the MEI( G, F ) pr oblem with ins aprx ( G, F ) cr ossings such that ins aprx ( G, F ) = ins( G, F ) + O ( ∆ k log k + k 2 ) , or mor e pr e cisely (2) ins aprx ( G, F ) ≤ ins( G, F ) + 2 k b log 2 k c ·  1 2 ∆  + 1 2  k 2 − k  . (3) Conse quently, this gives an appr oximate solution to the cr ossing numb er pr oblem cr aprx ( G + F ) = O ( ∆k ) · cr( G + F ) + O ( ∆ k log k + k 2 ) , or mor e pr e cisely (4) cr aprx ( G + F ) ≤ b 1 2 ∆ c · 2 k · cr( G + F ) + 2 k b log 2 k c ·  1 2 ∆  + 1 2  k 2 − k  . (5) Notice the constan t-factor approximation ratio when the degree of G and the size of F are b ounded. W e remark that the assumption of connectivity of G is necessary in the context of Theorem 1.2. F or disconnected G , the approximation guaran tee for ins aprx ( G, F ) would b e the same as for cr aprx ( G + F ) in (5). Concerning practical usability , our Algorithm 3.7, in fact, seems to b e the ﬁrst dir e ctly implementable and practically useful algorithm in this area, next to the 3 single edge insertion. In [6], an implementation of this algorithm is compared to the strongest known heuristics in practice, and oﬀers the arguably b est balance b et w een running time (b eing faster than most heuristics) and solution quality (second only to a v ery long running heuristic). 2 Decomp ositions and Embedding Preferences W e use the standard terminology of graph theory . By default, we use the term gr aph to refer to a lo opless multigraph. This means that w e allow p ar al lel e dges (but no lo ops), and when sp eaking ab out a cycle in a graph, we include also the case of a 2-cycle formed by a pair of parallel edges. If there is no danger of confusion b et w een parallel edges, we denote an edge with the ends u and v c hieﬂy by uv . W e pay particular atten tion to graph connectivity . A cut vertex in a connected graph G is a vertex u such that G − u is disconnected. A graph G is bic onne cte d if G has at least 3 vertices and no cut vertex, and G is tric onne cte d if G has at least 4 v ertices and no vertex-cut of size ≤ 2. A blo ck in a graph G is a maximal subgraph H ⊆ G such that H contains no cut vertex (of H ). In other words, a blo c k H is a maximal biconnected subgraph of G , or H has at most 2 vertices, whic h are not contained in any common larger blo c k. A dr awing of a graph G = ( V , E ) is a mapping of the vertices V to distinct p oin ts on a surface Σ , and of the edges E to simple curves on Σ , connecting their resp ectiv e end p oints but not con taining any other vertex p oin t. Unless explicitly sp eciﬁed, we will alw ays assume Σ to b e the plane (or, equiv alen tly , the sphere). A cr ossing is a common p oin t of tw o distinct edge curves, other than their common end p oint. Then, a dra wing is plane if there are no crossings. A graph is planar , if and only if it allo ws a plane drawing. The cr ossing numb er problem asks for a drawing of a given graph G with the least p ossible num b er cr( G ) of pairwise edge crossings (while there exists other deﬁnitions of a crossing num b er such as the pair or o dd crossing n umbers, those are not the sub ject of our pap er). By saying “pairwise edge crossings” we w ould lik e to emphasize that we count a crossing p oin t x separately for every pair of edges meeting in x (e.g., if k edges meet in x , then this accounts for  k 2  crossings). It is well established that the search for an optimal solution to the ab o v e crossing num ber problem can b e restricted to so called go o d drawings: an y pair of edges crosses at most once, adjacent edges do not cross, and there is no p oin t on Σ that is a crossing of three or more edges. A plane emb e dding G 0 of a connected graph G is G augmented with the cyclic orders of the edges around their inciden t vertices, suc h that there is a plane drawing of G resp ecting these orders. Embeddings hence form equiv alence classes ov er all plane drawings of G on the sphere. Cho osing any of the thereby induced fac es —the regions on the sphere enclosed by edge curves—as the (in- ﬁnite, unbounded) outer fac e gives an actual drawing in the plane. How ev er, it is technically easier to keep the “freedom of choice” of the outer face, and hence to work with plane drawings and embeddings on the sphere (unless we 4 explicitly sp ecify otherwise). Observ e that, when inv erting all the cyclic orders of an em b edding, we mirr or the em b edding, and consequently the corresp onding dra wings. In regard of Deﬁnition 1.1, if G F is a drawing of the graph G + F such that the restriction of G F to G giv es a plane em b edding G 0 , then w e c hieﬂy refer to G F as to G 0 + F . Giv en a plane embedding G 0 of G , w e deﬁne its dual G ∗ 0 as the em b edded graph that has a (dual) vertex for each face in G 0 ; dual vertices are joined by a (dual) edge for eac h (primal) edge shared b y their resp ectiv e (primal) faces. The cyclic order of the (dual) edges around any common incident (dual) vertex v ∗ , is induced by the cyclic order of the (primal) edges around the (primal) face corresp onding to v ∗ . W e may refer to a path in G ∗ 0 as to a dual p ath in G 0 . 2.1 Insertion problems and decomp osition trees When dealing with insertion problems, we alwa ys consider a connected planar graph G with a set F of additional k edges (with the ends in V ( G )) not present in E ( G ). Note that, since we allow m ultigraphs, an edge from F may b e parallel to an existing edge of G . Insertion algorithms typically work in tw o phases: ﬁrst, they choose an em- b edding G 0 of G ; then they ﬁx G 0 and draw the edges F within it. In this con text, we also use the following terminology . Deﬁnition 2.1 (Inserti on path). Consider a connected planar graph G and v 1 , v 2 ∈ V ( G ). Let G 0 b e a plane em b edding of G . An insertion p ath of { v 1 , v 2 } is a shortest dual path in G 0 from a face incident to v 1 to a face incident to v 2 .  Claim 2.2. L et G 0 b e a plane emb e dding of a gr aph and v 1 , v 2 ∈ V ( G 0 ) . A new e dge v 1 v 2 c an b e dr awn in G 0 with at most k cr ossings if, and only if, ther e is an insertion p ath of { v 1 , v 2 } in G 0 of length at most k . u t Our approach to edge insertion will use suitable tree-structured decomp osi- tions of the giv en planar graph, according to its connectivity . T he concept of these decomp ositions is also illustrated with an example in Figure 1. Deﬁnition 2.3 (BC-tree). Let G b e a connected graph. The BC-tr e e B = B ( G ) of G is a tree that satisﬁes the follo wing prop erties: i) B has t wo diﬀerent no de types: B- and C-no des. ii) F or every cut vertex in G , B con tains a unique corresp onding C-no de. iii) F or every blo ck in G , B contains a unique corresp onding B-no de. iv) No tw o B-, and no tw o C-no des are adjacent. A B-no de is adjacent to a C-no de iﬀ the corresp onding blo c k contains the corresp onding cut vertex.  T o further decomp ose the blo c ks, we consider SPQR-trees for each non-trivial B-no de (i.e., whose blo c k con tains more than tw o vertices). This decomp osition w as ﬁrst deﬁned in [12], based on prior w ork of [2, 24]. Even though more compli- cated than the BC-tree, it requires also only linear size and can b e constructed in linear time [16, 22]. W e are mainly interested in the prop ert y that an SPQR-tree 5 can b e used to eﬃciently represent and enumerate all (p otentially exp onen tially man y) plane embeddings of its underlying graph. F or conciseness, we call our tree SPR-tr e e , as we do not require no des of type Q. Deﬁnition 2.4 (SPR-t ree, cf. [5]). Let H b e a biconnected graph with at least three v ertices. The SPR-tr e e T of H is the (unique) smallest tree satisfying the follo wing prop erties: i) Eac h no de ν in T holds a sp eciﬁc (small) graph S ν = ( V ν , E ν ) where V ν ⊆ V ( H ), called a skeleton . Eac h edge f of E ν is either a r e al edge f ∈ E ( H ), or a virtual edge f = uv 6∈ E ( H ) (while still, u, v ∈ V ( H )). ii) T has three diﬀerent no de types with the following skeleton structures: S: the skeleton S ν is a cycle (of length 2 or more)—it represents a serial comp onen t; P: the sk eleton S ν consists of tw o vertices and at least three m ultiple edges b et w een them—it represents a p ar al lel comp onen t; R: the skeleton S ν is a simple triconnected graph on at least four vertices—it is “rigid” . iii) F or ev ery edge ν µ in T we ha ve | V ν ∩ V µ | = 2. These t wo common vertices, sa y x, y , form a vertex 2-cut (a split p air ) in H . Skeleton S ν con tains a sp eciﬁc virtual edge e µ ∈ E ( S ν ) that represents the node µ and, symmetrically , some sp eciﬁc e ν ∈ E ( S µ ) represen ts ν ; b oth e ν , e µ ha ve the ends x, y . These tw o virtual edges ma y refer to one another as twins . iv) The original graph H can b e obtained by recursively applying the follow- ing op eration of merging: F or an edge ν µ ∈ E ( T ), let e µ , e ν b e the twin pair of virtual edges as in (iii) connecting the same x, y . A mer ge d graph ( S ν ∪ S µ ) − { e µ , e ν } is obtained b y gluing the tw o skeletons together at x, y and remo ving e µ , e ν .  W e remark that SPQR-trees ha ve also b een used in the aforementioned [11], though with a diﬀeren t approach. W e use a sligh tly mo diﬁed v ersion of the SPR- tree, whic h “inserts” a degenerate S-no de b et w een each pair of P- or R-no des: Deﬁnition 2.5 (sSPR- tree). Let H b e a biconnected graph with at least three v ertices. A serialize d SPR-tr e e ( sSPR for short) T = T ( H ) of H is the unique smallest tree satisfying the prop erties (i)–(iv) of SPR-trees and additionally the follo wing; v) ev ery edge in T has precisely one end b eing an S-no de.  In traditional SPR-trees, S-no de skeletons will alwa ys contain at least three edges, due to their minimalit y . Because of prop ert y (v), sSPR-trees may , ho w- ev er, no w also con tain S-nodes represen ting 2-cycles. In fact, it is trivial to obtain an sSPR-tree from an SPR-tree by sub dividing any edge that it not incident to an S-no de with such a 2-cycle S-no de. W e observe that the sSPR-tree retains es- sen tially all prop erties of SPR-trees, in particular it also has only linear size, can b e computed in linear time, and all the previously known insertion algorithms 6 Giv en graph: BC-tree: Con-tree: S S P C C C S D S S P R S P S R S S Decomp osition graph: S S P C C C S D S S P R S P S R S S SPR-tree & sk eletons of largest blo c k: R S P R S S sSPR-tree & sk eletons of largest blo c k: R S P S R S S Fig. 1. Example for the v arious decomp ositions. 7 (most imp ortan tly the single edge insertion algorithm [18]) can b e p erformed using the sSPR-tree without an y mo diﬁcations. W e are particularly interested in the amalgamated version of the ab o ve de- comp ositions, chieﬂy denoted by c on-tr e e : Deﬁnition 2.6 (Con-tree). Given a connected, planar graph G , let the c on- tr e e C = C ( G ) b e formed of the BC-tree B ( G ) that holds sSPR-trees T ( H ) for all non-trivial blo c ks H of G . F or technical reasons, if H is a trivial 2-vertex blo ck, w e set T ( H ) to b e the tree formed by a single dumm y no de, called a D-no de , whose skeleton is H . 3 Also, let the skeleton S ν of a C-no de ν simply b e the corresp onding cut vertex.  Clearly , the linear-sized con-tree C ( G ) can b e obtained from G in linear time. F urthermore, we will sometimes treat this tw o-level con-tree as follows: Deﬁnition 2.7 (Fl attening a con-tree, decomp osition graph). In the set- ting of Deﬁnition 2.6, let a de c omp osition gr aph D ( G ) of G b e constructed from the union of all the sSPR trees T ( H ) o ver the blo c ks H of G and of all the C-no des of B ( G ), as follows: i) Observ e that for each blo c k H in G holding a cut vertex x of G , there is one or possibly multiple no des of T ( H ) whose skeletons contain x , and not all of them are P-no des. Call these no des of T ( H ) whose skeletons contain x the mates of x from T ( H ). ii) In D ( G ), for every C-node µ holding a cut vertex x of G and for every blo c k H of G con taining x , mak e µ adjac ent to all the mates of x from T ( H ) that are not P-no des.  This “ﬂat” view of a decomp osition graph will later b e implicitly used, for in- stance, when deﬁning a con-path (Deﬁnition 2.9). The in tentional exclusion of P-no des in (ii) is of technical nature and will b e justiﬁed later in Claim 2.10. T o summarize, a con-tree (as well as the related decomp osition graph) pro- vides us with a natural and easy wa y to encode all embeddings of a given planar graph. An S-no de sk eleton has a unique embedding. F or an R-no de, we can c ho ose b et ween the unique embedding of its skeleton and the mirror image. F or a P-no de, w e can choose an arbitrary cyclic p ermutation of the skeleton edges. F or a C-no de, a precise description of the choice is more tricky , but it roughly means we can c ho ose which face of each blo c k (assuming the blo c ks are already em b edded) holds which other blo c k. On the other hand, given a particular plane em b edding of a graph G , it is trivial to deduce the em beddings of all the skeletons within the con-tree C ( G )—we shall formalize this observ ation in Deﬁnition 2.12 and Claim 2.13. 3 In a simple graph, a trivial blo c k is alw ays a bridge in G . In m ultigraphs, H may also b e a set of parallel edges betw een a vertex pair x, y , whose remov al would disconnect x, y . In the latter case, it is trivial that the precise order of the edges is irrelev ant, and w e could represent H via a single edge e H and its multipli city | E ( H ) | instead. 8 2.2 Con-c hains and con-paths In this paper we refer to the optimal single-edge insertion algorithm by Gutw enger et al. [18]. Consider (cf. Deﬁnition 1.1 for k = 1) a connected planar graph G and let v 1 , v 2 b e the vertices we wan t to connect by a new edge. In a nutshell (see Section 2.5 for details), the algorithm of [18] sho ws that an optimal plane em b edding (of G ) for inserting v 1 v 2 in to G dep ends only on the con-tree no des “b et w een” some no de containing v 1 and one containing v 2 . In this regard, w e bring the follo wing sp ecialized deﬁnitions. Deﬁnition 2.8 (Con-chain). Consider a connected planar graph G , and its con-tree C ( G ). Let v 1 , v 2 ∈ V ( G ). The c on-chain Q v 1 v 2 of the pair { v 1 , v 2 } in C ( G ) can b e uniquely deﬁned as consisting of the following: (i) the unique shortest path Q in B ( G ) b et w een a B-no de β 1 whose blo c k con- tains v 1 and a B-no de β 2 whose blo c k con tains v 2 ; (ii) for each B-no de β along Q , tw o b or der vertic es w β 1 , w β 2 : for i = 1 , 2, if β = β i let w β i = v i ; otherwise let w β i b e the cut vertex in G corresp onding to the (unique) C-no de neighbor of β closest to β i ; 4 and (iii) again for each β on Q , the no de sequence Q β of the unique shortest path in T ( H β ) connecting no des of T ( H β ) whose skeletons contain the b order v ertices w β 1 and w β 2 in this order. In (iii), uniqueness of the path generally follo ws from the fact that T ( H β ) is a tree. F or the p ossible degenerate case that Q β consists of a single no de (whose sk eleton holds b oth w β 1 , w β 2 ) but is not unique, we choose the “ﬁrst” such no de, according to some arbitrary but ﬁxed order of the decomposition no des. (In fact, this sp ecial case is show cased in Figure 1.)  Deﬁnition 2.9 (Con-path). In the setting of Deﬁnition 2.8, a c on-p ath P v 1 ,v 2 := P ( Q v 1 ,v 2 ) of { v 1 , v 2 } in C ( G ) is the sequence of decomp osition no des (of type S, P , R, C, and D) resulting from Q (which contains B- and C-no des), b y replacing each B-no de β by its corresp onding sequence Q β .  Ob viously , no con-path may start or end with a C-no de. Moreov er: Claim 2.10. A c on-chain pie c e Q β (cf. Def. 2.8(iii)) c an neither start nor end with a P-no de. Ther efor e, a P-no de c an o c cur in a c on-p ath only b etwe en two S-no des, and never at the end nor adjac ent to a C-no de. u t The term c on-p ath is justiﬁed by the fact that this no de sequence is a path in the decomp osition graph D ( G ) from Deﬁnition 2.7. W e may consider a con-path implicitly orien ted, when suitable. Ev en though a decomp osition graph D ( G ) is not a tree, our con-paths in D ( G ) “b eha v e” similarly as paths do in a tree: 4 Note that w β 2 = w β 0 1 if β , β 0 are consecutiv e B-no des on Q in this order. 9 Claim 2.11. Consider two c on-p aths P 1 , P 2 of p airs { v 1 , w 1 } and { v 2 , w 2 } in the de c omp osition gr aph D ( G ) of a gr aph G . Then P 1 ∩ P 2 , if non-empty, forms a c onse cutive subse quenc e in e ach of P 1 , P 2 . Pr o of. In this pro of we view P 1 , P 2 as paths in the graph D ( G ). Assuming a con tradiction, there exist distinct subpaths P 0 i ⊆ P i , i = 1 , 2, such that P 0 1 , P 0 2 share common ends α, β ∈ V ( D ) but are internally disjoint otherwise. Since C- no des are cut no des in D ( G ) by Deﬁnition 2.7, P 0 1 ∪ P 0 2 con tains no C-no des except p ossibly α and/or β . If neither of α, β is a C-no de, then P 0 1 , P 0 2 are contained in the same sSPR-tree of C ( G ) and so P 0 1 = P 0 2 , a contradiction. Similarly , if b oth α, β are C-no des then b y uniqueness claimed in Deﬁnition 2.8.(iii), it is P 0 1 = P 0 2 . It remains to consider that, up to symmetry , α is a C-no de and β b elongs to an sSPR-tree T ( H ) where H ⊆ G is a blo c k incident with a ∈ V ( G ), the cut vertex of α . Let α i b e the neighbor of α on P 0 i , i = 1 , 2. Then β lies on the unique α 1 - α 2 path in T ( H ), and since b oth the skeletons S α 1 , S α 2 con tain a , it is a ∈ V ( S β ). Unless β is a P-no de, P 0 1 = P 0 2 is thus formed by a single edge αβ . In the case of β b eing a P-no de, cf. Claim 2.10, β is not an end of either P 1 , P 2 . Therefore, if β i ∈ V ( P i ) \ V ( P 0 i ) is the (other) neighbor of β on P i , i = 1 , 2, w e hav e a ∈ V ( S β i ) and by minimality in Deﬁnition 2.8.(iii) it is β 6∈ V ( P i ), a con tradiction again. u t 2.3 T ow ards “embedding preferences” On a high level, the starting p oin t of our algorithm is to solv e the single edge insertion problem for each edge { v 1 , v 2 } ∈ F into G indep enden tly . In this w a y we obtain so called emb e dding pr efer enc es (Deﬁnition 2.15) for eac h node ν ∈ P v 1 ,v 2 . Equipp ed with all these preferences for all edges of F , we hav e to decide on an em b edding G 0 of G which, essentially , “honors” as many of these preferences as p ossible. It should be understoo d that the preferences for distinct edges of F ma y naturally conﬂict with one another. W e will ensure that we pick an embedding G 0 whic h guarantees that any { v 1 , v 2 } ∈ F can b e inserted into G 0 (as a new edge v 1 v 2 ) without “to o many” additional crossings, compared to its individual optim um ins( G, v 1 v 2 ). The exact scheme for our em b edding preferences is a critical part of the proof of our algorithm, and in order to explain our complete d eﬁnitions (Section 2.4), it seems worth while to discuss the situation and drawbac ks of p ossible alternativ es b eforehand. W e start with a summary of the algorithm [18] to optimally insert a single edge. Our description is v ery informal since we nev er use the details of this algorithm in our pap er, but the summary will help us to explain the (somehow unexp ected) deep technical problems connected with our extension to multiple edges. Single e dge insertion algorithm [18]. W e can interpret the algorithmic steps of [18] for ins( G, v 1 v 2 ) as what w e call simple embedding preferences. In our terminology of Deﬁnition 2.8, the insertion algorithm for the edge v 1 v 2 ﬁrst con- siders individually each B-no de β of Q v 1 ,v 2 . Let Q β = ( ν 1 , ν 2 , . . . , ν r ) b e ordered 10 suc h that ν 1 con tains w β 1 and ν r con tains w β 2 . F or ν i ∈ Q β , one indep enden tly computes an em b edding and a lo cal insertion path as follows. The source (target) of the lo cal insertion path is either the v ertex w β 1 ( w β 2 ), if i = 1 ( i = r ), or otherwise e ν i − 1 ( e ν i +1 ) in S ν i —the virtual edge(s) corresponding to the predecessor and the successor of ν i , resp ectiv ely . The insertion path is computed in the dual of the skeleton S ν i : F or S-no des, the path is simply one of the tw o faces, and no crossings arise. Similarly , no crossings arise for P-no des; one chooses a skeleton em b edding in whic h the source and the target lie on a common face. This face then constitutes the insertion path. In case of an R-node, the embedding of S ν i is ﬁxed and one chooses an y shortest path in its dual as the insertion path. Crossins ma y , unav oidably , arise in this case. In eac h of the cases, if the source or the target is a virtual edge, one also notes whether the insertion path attaches to it from the left or the right side (left/righ t with resp ect to an arbitrarily ﬁxed but consistent direction of all the virtual edges). This solution and note together constitute the simple emb e dding pr efer enc e for each no de of Q β in the con-tree of G . All the simple embedding preferences along Q β can b e simultaneously sat- isﬁed, by stepwise “gluing together” the individual sub em b eddings along twin pairs of virtual edges: this is trivial when the lo cal insertion paths of adjacent no des attac h to the t win pair from diﬀerent sides, and one ﬂips the next skeleton em b edding b efore gluing in the other case. F or other no des without preferences, one picks any em b edding, as the insertion solution is indep endent of this decision. Ha ving embedded the blo c ks of G , it is no w easy to deal with C-no des. F or any C-no de γ of Q v 1 ,v 2 (corresp onding to a cut vertex x ∈ V ( G )), let H , H 0 ⊆ G b e the inciden t blo c ks. The corresp onding “lo ose ends” of the previously computed insertion paths in H and H 0 attac h to x from certain faces ϕ, ϕ 0 of em b edded H , H 0 , resp ectively . In an embedding of the union H ∪ H 0 one identiﬁes ϕ with ϕ 0 , suc h that the insertion paths are joined together without additional crossings. All remaining blo c ks can b e joined arbitrarily as the insertion solution is indep enden t of these decisions. Pr op erties of go o d pr efer enc e schemes. In our case of multiple edges, the ab o ve description has inherent problems, some rather ob vious, some more subtle. First, simple prefereces require a stepwise resolution (i.e., having to decide all the blo c ks, b efore being able to prop erly deﬁne C -node preferences), which do es not allo w for an easy wa y of combining multiple insertion paths. Second, simple preferences are capable of handling only a quite restricted class of embeddings. In our case, this would inevitably lead to a situation in whic h tw o distinct insertion paths set diﬀerent preferences for the no des they follo w together, even though there exists an equally optimal solution for one of the paths that nearly matches the preferences of the other path. A very simple example of this is when the ﬁrst path decides for an embedding G 0 while the second one decides for the mirror image of G 0 . Note, ho wev er, that there is no big problem with routing through rigid skeletons of R-no des, as one can alw ays deﬁne a unique canonical order on all the dual paths there, and a unique default c hoice b et ween the skeleton and its mirror image (ﬂip). 11 The aforemention problem gets tougher when considering S- and P-no des (and also C-no des in greater generality). Consider that an embedding of a P- no de ν skeleton is sp eciﬁed by a clo c kwise order of its edges (yes, it is actually enough to sp ecify a pair of consecutive edges). How ever, another sp eciﬁcation at ν may reverse the edge order of the ν skeleton and simultaneously ﬂip b oth the adjacen t sk eleton em beddings. This results in, essentially , the same embedding of G , and has to be captured as “the same” b y our new preference sc heme. But then w e w ould mix the concepts of sp ecifying sub em b eddings (e.g., the edge order at P-no des) and of handling the gluing op eration (ﬂips). In this p erspective, our extension from SPR- to sSPR-trees can be seen as a ﬁrst step to decouple ﬂipping decisions from ordering decisions within P-no des—it allows us to enco de, at a mandatory S-no de, whether the adjacen t skeletons are ﬂipp ed “the same wa y” or “the other w ay”; with sp ecial lab els nonswitching / switching . Though, the problems are not ov er yet. One would also, naturally , lik e to deﬁne the preferences so that it is trivial to v alidate if some given embedding satisﬁes the preferences or not. In other words, one would like that every bit of information stored as a preference has a directly observ able counterpart in the ﬁnal embedding G 0 . While there are wa ys to rigorously deﬁne a suitable prefer- ence sc heme ac hieving this goal, 5 the solution is not satisfactory for the following reason. With any directly observ able preference scheme, it seems inevitable that v alidation of the embedding preference of one no de requires us to lo ok at the em b edding preferences of some of its neighbors (sometimes up to distance tw o). Not surprisingly , this would bring big complications for the pro ofs, resulting in long (and diﬃcult to v erify) case-chec king arguments. In our new preference scheme, w e hence trade the (part of ) direct observ abil- it y prop ert y for a new prop erty of str ong lo c ality —demanding that v alidation of the embedding preference of one no de is p ossible without lo oking at the pref- erences of an y other no des. F or this purp ose we shall introduce a new kind of “unobserv able” information in to an em b edding—a so called spin (of virtual edges), which helps to achiev e the goal of strong lo calit y b y providing us with kind of a “comm unication channel” b etw een neighboring preferences. All of this is the sub ject of the following rather technical Section 2.4. 2.4 Em b edding preferences in a Con-tree In this section we give the formal deﬁnitions establishing our scheme of em- b edding preferences for con-chains that achiev es the goal of strong lo calit y , as informally explained in Section 2.3. W e start with formally sp ecifying what an em b edding is (Deﬁnition 2.12) and what the partially unobserv able information w e add to it is (Deﬁnition 2.14 and Claim 2.17). Then, w e deﬁne the exact pieces of information w e are going to store at a con-c hain no de as its embedding preference (Deﬁnition 2.15), and w e couple this deﬁnition with a sp eciﬁcation (resp ecting strong lo calit y) of what 5 F or example, it is the preference sc heme originally described in the conference version of this research [8]. 12 it means that a certain collection of em b edding preferences is honored by an em b edding of G (Deﬁnition 2.16). F or reference in the follo wing deﬁnitions, w e need to deﬁne certain “defaults” for em b eddings of a graph G . Let G d b e an arbitrary but ﬁxe d embedding of G in the plane (i.e., including the sp eciﬁcation of the unbounded face). F rom G d w e deriv e (a) the default emb e dding for each R-no de skeleton (distinguishable from its mirror), and (b) the default fac e for each S-no de skeleton (distinguishable from its other face) by pic king the b ounded face of this skeleton in G d . Note that the default face is w ell-deﬁned even for an S-no de sk eleton forming a 2- cycle since the skeleton edges are lab eled. Sp ecially , for each D-no de skeleton (a bunch of parallel non-virtual edges), w e observe that it is alwa ys p ossible to dra w all its edges close together such that the rest of G is em b edded in one face of it and that no crossings with the inserted edges arise. This chosen face, in an y embedding of G , will b e called the default fac e of a D-no de skeleton (while disregarding the names and order of the sk eleton edges). Deﬁnition 2.12 (Embedding sp eciﬁcation). Consider a connected planar graph G , an embedding G 1 of G , and the con-tree C := C ( G ). Let an emb e dding sp e ciﬁc ation E ( G 1 ) b e a collection of information sp eciﬁed as follows: i) F or eac h R-no de ν of C , we sp ecify one of the lab els flipped or nonflipped ; the information is nonflipped exactly when the em b edding of S ν induced b y G 1 is the default one. ii) F or eac h P-no de ν of C , we sp ecify (an arbitrary) one of the tw o vertices of V ( S ν ) and the cyclic clo c kwise order of the edges of S ν around it in the em b edding of S ν induced b y G 1 . iii) F or eac h C-no de ν of C and each pair H 1 , H 2 ⊆ G 1 of incident blo c ks, we sp ecify the face of H j that con tains H 3 − j in the em b edding G 1 , for j = 1 , 2. An embedding G 1 uniquely determines the embedding sp eciﬁcation E ( G 1 ) by deﬁnition. Con versely , it immediately follows that: Claim 2.13. If G 1 , G 2 ar e two emb e ddings of a c onne cte d planar gr aph G such that E ( G 1 ) = E ( G 2 ) , then G 1 and G 2 ar e e quivalent (and also not mirr or e d images of e ach other). u t While this embedding sp e ciﬁc ation of Deﬁnition 2.12 cannot be used directly as a basis for our em b edding pr efer enc e scheme, it serves as a blue print of what our preferences need to mo del. Recall our discussion from Section 2.3 ab out ﬂipping skeleton embeddings during the gluing op eration of adjacent skeletons. F or the intended purp ose of decoupling the ﬂipping decision of a glue op eration of Deﬁnition 2.4 from the em b edding decision for sk eletons, we introduce the concept of spins in an embed- ding. In terestingly , in general, spin v alues are not fully determined by a particular em b edding. A spin can also b e (informally) seen as a 1-bit communication c han- nel transferring certain information b etw een nonadjacent con-tree no des, which will allo w us to establish the desired strong lo cality of em b edding preferences. 13 Deﬁnition 2.14 (Enriched embedding, spins) . Consider a connected pla- nar graph G , its embedding G 1 and the con-tree C ( G ). A CS-p air is a tuple ( c, ν ) where c is a cut vertex in G and ν is an S-no de of C ( G ) whose skeleton con tains c . In an enriche d em b edding ˆ G 1 of G 1 , we charge each virtual edge and each CS-pair of the con-tree C ( G ) with either a p ositive or a ne gative spin , sub ject to the following prop erties: (E1) A virtual edge and its t win will alwa ys b e charged identically . (E2) F or an R-no de % , if the embedding of the skeleton S % is the default one (i.e., % is lab eled nonflipped in E ( G 1 )), then all the virtual edges in S % are charged with a p ositiv e spin; otherwise, all those virtual edges are c harged with a negative spin.  The role of CS-pairs ( c, ν ) deserv es a further informal explanation. First, note that c in such a pair is a mate of ν in the sense of Deﬁnition 2.7. Second, a CS- pair will b e used conceptually similarly to a pair of twin virtual edges of an sSPR tree—while twin pairs of virtual edges are used to glue pieces of an sSPR tree together, CS-pairs will b e used to specify the precise wa y t w o adjacen t blo c ks are glued together at a cut vertex. Though, we will need this additional information only in the case when ν is an S-no de (and not for R-no des for which we may explicitly refer to a certain face). Since R-no des are never directly adjacent, there cannot arise any conﬂicting spin v alues from Deﬁnition 2.14. Most imp ortan tly , all spin v alues not predeter- minded by prop erties E1 and E2 are only lo osely correlated with the embedding G 1 , and they will not b e part of the em b edding preference. Their role is to func- tion as a mediator betw een our embedding preferences and an actual embedding. In other words, we will only ask whether ther e exist spin v alues such that an em b edding satisﬁes our preferences, and not what the spin v alues are. The following tw o entangled deﬁnitions achiev e all our goals concerning em- b edding preferences in a con-tree and their interpretation. Deﬁnition 2.15 (No de embedding preferences for a con-chain). Consider a connected planar graph G , its con-tree C ( G ), and the decomp osition graph D := D ( G ) (Deﬁnition 2.7). An emb e dding pr efer enc e π ν of a no de ν ∈ V ( D ) is deﬁned (only) if ν is an S-, P-, or C-no de. One piece of information that π ν stores is a pair of distinct no des µ 1 , µ 2 ∈ V ( D ), called the π ν -p e ers of ν , such that µ 1 , µ 2 are neigh b ors of ν in D . F urthermore: (P1) If ν is a P-no de, w e only store these p eers µ 1 , µ 2 (whic h, in this case, are S-no des by deﬁnition) in π ν . (P2) If ν is an S-no de, w e additionally store in π ν one of the labels switching or nonswitching . Ho w ever, if neither µ 1 nor µ 2 is an R-node, the stored lab el must alwa ys b e nonswitching . (P3) If ν is an C-node, then the stored π ν -p eers µ 1 , µ 2 are R-, D- or S-no des by deﬁnition. F or i = 1 , 2, if the p eer µ i is an R-no de, then we additionally store in π ν a lab el sp ecifying a face in the skeleton S µ i . 14 Consider a non-adjacent vertex pair v 1 , v 2 ∈ V ( G ) and its con-chain Q v 1 ,v 2 in C ( G ). Let P := P ( Q v 1 ,v 2 ) ⊆ D b e the corresp onding con-path. An emb e dding pr efer enc e Π v 1 v 2 of Q v 1 ,v 2 (equiv alently , a pr efer enc e of P ) is a collection of em b edding preferences π ν o ver those internal no des ν of P that are neither R- nor D-no des, sub ject to the following restriction: (P4) F or every individual preference π ν in Π v 1 v 2 , the t wo π ν -p eers of ν must b e the tw o neighbors of ν on P .  F or consistency , we sa y that a no de ν whose embedding preference is not deﬁned in Deﬁnition 2.15 ( ν migh t b e an R- or D-no de), has a void em b edding preference. Then, we can rigorously sp eak ab out embedding preferences of al l the internal no des of a con-path P , or of all no des in an arbitrary subset of V ( D ). Deﬁnition 2.16 (Honoring an embedding preference). Consider a connected planar graph G and any embedding G 1 of G , sp eciﬁed via its embedding sp eciﬁcation E := E ( G 1 ). Let U b e a subset of no des of C ( G ) and Π U b e a collection of (arbitrary) no de embedding preferences π ν o ver ν ∈ U . W e say that G 1 honors the emb e dding pr efer enc es Π U if there exists an enriched em b edding ˆ G 1 of G 1 suc h that, for ev ery ν ∈ U where π ν is non-void, ˆ G 1 is go od for π ν . That is, ˆ G 1 is go o d for π ν (where ν falls into one of the cases (P1)–(P3) of Deﬁnition 2.15) if the appropriate one of the following cases holds: case (P1): ν is a P-no de. Let µ 1 , µ 2 b e the π ν -p eers (b oth b eing S-no des) of ν , and let e µ 1 , e µ 2 denote the virtual edges in S ν corresp onding to µ 1 , µ 2 , resp ectiv ely . The cyclic order of the edges in S ν sp eciﬁed in E is such that e µ 1 and e µ 2 o ccur as neighbors—they form a common face ϕ in S ν . Moreov er, for each i = 1 , 2, the face ϕ corresp onds to the default face of S µ i if e µ i has the p ositiv e spin in ˆ G 1 , and ϕ corresp onds to the other face of S µ i if the spin is negativ e. case (P2): ν is an S-no de. There are tw o spin v alues asso ciated with π ν via ˆ G 1 : for i = 1 , 2, the i -th spin v alue is the one charged to the virtual edge e µ i if the π ν -p eer µ i is an R- or P-no de, and otherwise the i -th spin v alue is the one charged to the CS-pair formed by ν and the cut vertex of the C-no de µ i . If the lab el in π ν is switching then exactly one of these spin v alues of ˆ G 1 is p ositiv e and the other one negative, while if the lab el is nonswitching then b oth these spin v alues hav e to b e the same. 6 case (P3): ν is a C-no de. Let H 1 , H 2 ⊆ G b e the blo cks the π ν -p eers µ 1 , µ 2 of ν b elong to, and let c b e the cut vertex corresp onding to ν . There are tw o sk eleton faces, ϕ 1 of S µ 1 and ϕ 2 of S µ 2 , asso ciated with π ν via ˆ G 1 as follows. Let i ∈ { 1 , 2 } . If µ i is an R-no de then ϕ i is explicitly given as a lab el in π ν . If µ i is a D-no de then ϕ i is the default face of S µ i . If µ i is an S-no de, then ϕ i is the default face of S µ i in the case that the CS-pair ( c, µ i ) is c harged with a p ositiv e spin, and otherwise (negative spin) ϕ i is the other face of 6 Note that (some of ) the spin v alues in ˆ G 1 are correlated with the speciﬁcation E according to Deﬁnition 2.14, and so E is implicitly considered in the case (P2), to o. 15 S µ i . The t wo faces sp eciﬁed in E for H 1 , H 2 at ν must corresp ond to ϕ 1 and ϕ 2 (disregarding their orien tation), resp ectiv ely .  V oid em b edding preferences from Π U , and possible preferences of other nodes not in U , do not matter when deciding whether an embedding G 1 honors Π U as ab o ve. W e also call an y enriched embedding ˆ G 1 of G 1 that satisﬁes all the requiremen ts of Deﬁnition 2.16 a go o d enriche d emb e dding for Π U . Claim 2.17. Assume that a plane emb e dding G 1 of G honors emb e dding pr ef- er enc es Π := Π v 1 v 2 of a c on-chain Q v 1 ,v 2 . Then, the spin values char ge d to the virtual e dges and the CS-p airs along Q v 1 ,v 2 ar e c onsistent over al l p ossible go o d enriche d emb e ddings for Π . These spin values c an b e derive d dir e ctly fr om the emb e dding sp e ciﬁc ation E ( G 1 ) (w.r.t. the implicit default emb e dding G d of G ). Pr o of. Since G 1 honors Π , there exists at least one feasible spin assignment by deﬁnition. W e are going to show that the spin v alues are uniquely determined along the con-path P := P ( Q v 1 ,v 2 ). Consider any edge ν µ ∈ E ( P ). If one of ν, µ is an R-no de (and the other one is not a C-no de), then the tw o virtual edges asso ciated with ν µ in C ( G ) are c harged a spin v alue that is uniquely determined by Deﬁnition 2.14. Assume that, say , ν is a C-no de of a cut vertex c . Then a spin is c harged to the CS-pair ( c, µ ) only if µ is an S-no de. W e determine the spin v alue of ( c, µ ) by comparing the face sp eciﬁed in E ( G 1 ) for ν and the blo c k of S µ with the default face of S µ , as claimed in Deﬁnition 2.16, case (P3). The remaining case to consider is, up to symmetry , that ν is a P-no de and µ an S-no de. The spin v alue of the twin virtual edges e µ , e ν (in S ν , S µ , resp ectiv ely) is again determined by comparing the embedding of the P-skeleton S ν in E ( G 1 ) (precisely , its face sp eciﬁed by the preference π ν ) to the default face of S µ , as claimed in Deﬁnition 2.16, case (P1). u t The consequence of the ab o v e claim is very interesting. On the one hand, it tells us that, eﬀectively , we do not need to care to o m uch about enriched em b ed- dings from no w on, except for in “lo w-lev el” pro ofs. On the other han d, how ever, it is not straigh tforwardly p ossible to remo v e the spins from the deﬁnition of em- b edding preferences altogether. A key problem o ccurs with an S-no de ν and its virtual edge e µ suc h that an em b edding preference of the neigh b or µ do es not refer to ν (but to other no des): then, the spin v alue of e µ is undetermined and represen ts a “communication c hannel” b et w een the embedding preferences of adjacen t no des. Abandoning this vital “comm unication link” w ould require us, e.g., for a P- or C-no de in Deﬁnition 2.16, to refer also to the embedding pref- erences of the neighbors and thus would violate our goal of strong lo cality (with some nast y consequences in the later pro ofs). 2.5 Single edge insertion and embedding preferences Note that it is not a priori clear, given the additional conditions in Deﬁnitions 2.15, 2.16 and the elusive nature of spins, that our treatment of em b edding pref- erences is capable of rigorously describing ev ery optimal solution to the single 16 edge insertion problem. It is the task of coming Lemma 2.18 to pro ve this im- p ortan t and nontrivial ﬁnding. (While p erhaps lo oking similar to Claim 2.17, this task is in fact v ery diﬀerent from the former one.) Lemma 2.18. L et G b e a c onne cte d planar gr aph and v 1 , v 2 ∈ V ( G ) . Assume that G 0 is a plane emb e dding of G such that a new e dge f = v 1 v 2 c an b e dr awn into G 0 with ins( G, f ) cr ossings. Then ther e exists an emb e dding pr ef- er enc e Π v 1 v 2 of the c on-chain Q v 1 ,v 2 such that G 0 honors Π v 1 v 2 . Pr o of. Let P := P ( Q v 1 ,v 2 ) b e the corresp onding con-path, and E = E ( G 0 ) b e the em b edding sp eciﬁcation of G 0 . There are t wo tasks in this pro of: a) to deduce individual embedding preferences π ν (forming Π v 1 v 2 ) for all inter- nal no des of P , and b) to sp ecify an enriched embedding ˆ G 0 suc h that all the conditions of Deﬁni- tion 2.16 are satisﬁed for Π v 1 v 2 via ˆ G 0 . As for a), we easily determine all information except the switching attributes of the internal S-no des on P from Deﬁnitions 2.15, 2.16: the π ν -p eers are the neigh b ors of ν on P and, in the case of a C-no de ν with an R-neighbor, the face lab el(s) follows Deﬁnition 2.16, case (P3). Assuming for a momen t that, for each in ternal P-node ν of P , the tw o virtual edges of the π ν -p eers form a face in E , in b) we can determine spin v alues along P exactly as in the pro of of Claim 2.17. Remaining spin v alues c an b e charged arbitrarily , giving an enriched embedding ˆ G 0 of G 0 . Returning back to a), the switc hing attribute of each π ν where ν is an S-no de now follows from c harged spin v alues and Deﬁnition 2.16, case (P2). It remains to verify three facts; our assumption ab out P-no des of P , and fulﬁllmen t of Deﬁnition 2.15 (P2) and of Deﬁnition 2.16. This can b e done con- v eniently along the cases of the latter deﬁnition. Let ν b e an internal no de of P and µ 1 , µ 2 b e the tw o π ν -p eers of ν (its neighbors on P ). case (P1): ν is a P-node. F or i = 1 , 2, let H i ⊆ G b e the subgraph obtained b y recursiv ely gluing skeletons to the virtual edges of S µ i except to e ν . Up to symmetry b et ween the indices i = 1 , 2, either v i ∈ V ( H i ) or v i is separated from V ( S µ i ) ∪ { v 3 − i } b y a cut v ertex c of G such that c ∈ V ( H i ). Notice that V ( H 1 ) ∩ V ( H 2 ) = { p 1 , p 2 } is a split pair in G formed by V ( S ν ) = { p 1 , p 2 } . Ob viously , the edge f in optimal G 0 + f do es not cross any comp onen t represen ted by a virtual edge of S ν , other than p ossibly H 1 , H 2 . Therefore, the virtual edges e µ 1 , e µ 2 indeed form a face in the embedding speciﬁcation E of G 0 . case (P2): ν is an S-no de. The switc hing lab el of π ν has already b een chosen to satisfy Deﬁnition 2.16 via ˆ G 0 . As in the previous case, the edge f in optimal G 0 + f obviously do es not cross any of the comp onen ts represented b y the virtual edges of S ν other than p ossibly e µ 1 , e µ 2 . Hence, unless one of µ 1 , µ 2 is an R-no de (in which case e µ 1 or e µ 2 has b een charged a spin by Deﬁnition 2.14), b oth the spin v alues c harged for µ 1 and µ 2 (eac h may b e of a virtual edge e µ i or of a CS-pair ( x i , ν )) refer to the same face of S ν and the lab el of π ν is nonswitching , in agreemen t with Deﬁnition 2.15. 17 case (P3): ν is a C-no de. In this case there is nothing more to discuss since π ν and ˆ G 0 ha ve already b een chosen to satisfy Deﬁnition 2.16. u t In the algorithmic context, the k ey is to ﬁnd suc h an embedding preference Π v 1 v 2 as in Lemma 2.18 eﬃciently . Using Lemma 2.18, we translate the main result of [18] in to our slightly diﬀerent setting as follows: Theorem 2.19 (Gutw enger et al. [18], reform ulated). L et G b e a c on- ne cte d planar gr aph and v 1 , v 2 ∈ V ( G ) . L et Q v 1 ,v 2 b e the c on-chain of the p air { v 1 , v 2 } in C ( G ) . We c an ﬁnd, in line ar time, an emb e dding pr efer enc e Π v 1 v 2 of Q v 1 ,v 2 such that we r e quir e exactly ins( G, f ) cr ossings to dr aw a new e dge f = v 1 v 2 into any plane emb e dding G 0 of G that honors Π v 1 v 2 . An y em b edding preference Π v 1 v 2 with the prop erties as describ ed in Theo- rem 2.19 will b e called an optimal emb e dding pr efer enc e of (the con-chain of ) the pair { v 1 , v 2 } in G . Note that such an optimal preference Π v 1 v 2 is not neces- sarily unique and, furthermore, that optimality of inserting f do es not dep end on embedding sp eciﬁcations/preferences at the con-tree no des other than the in ternal no des of Q v 1 ,v 2 . 3 MEI Appro ximation Algorithm W e now ﬁnally hav e all the technical ingredients to discuss our new approxi- mation algorithm for the MEI( G, F ) problem (Deﬁnition 1.1). The ov erall idea of the approximation is to suitably combine the individually computed optimal em b edding preferences ov er all the edges of F , and to prov e that not to o many conﬂicts arise, leading to only few additional crossings as compared to the sum of the individual optimal solutions. The aforemen tioned strong lo calit y , achiev ed in Deﬁnition 2.16, will b e crucial for the algorithm. 3.1 Coherence, and repairing insertion paths Before jumping into a solution of the MEI problem, we hav e to analyze the kind of conﬂicts that arise b etw een optimal embedding preferences of tw o distinct pairs { v 1 , v 2 } and { w 1 , w 2 } from F . Intuitiv ely , if the con-chains of { v 1 , v 2 } and { w 1 , w 2 } lo cally trav erse the con-tree of G in the same wa y , then b oth of them should ha ve “the same” optimal individual embedding preferences there. Though, the formal ﬁnding is not as straightforw ard as the previous claim due to tw o small complications; ﬁrst, the situation of R-no des is more delicate (cf. Deﬁnition 3.1), and second, an optimal em b edding preference of a pair is not necessarily unique and this has to b e tak en into an account (cf. Lemma 3.2). Recall that our con-paths can b e view ed as paths in the decomp osition graph, and this will b e our default view in this section. Deﬁnition 3.1 (Coherence in con-paths). Consider tw o con-paths P , P 0 in the decomp osition graph of a connected planar graph G , and their non-empt y 18 in tersection R = P ∩ P 0 (whic h is a subpath by Claim 2.11). W e say that P , P 0 are c oher ent at ν ∈ V ( R ) if ν is an inner no de of R and the following holds: if µ 1 , µ 2 are the t wo neighbors of ν on R and µ i , i ∈ { 1 , 2 } , is an R-no de, then also µ i is an inner no de of R .  W e would like to add a small remark on the situation in whic h µ i is, sim ulta- neously , an end of R and an end of b oth P , P 0 . Assume the insertion edges f , f 0 deﬁning P , P 0 end in a common vertex of the skeleton of µ i . Then, we c ould also deﬁne the con-paths to b e “coherent” at ν . How ever, we refrain from doing so as it w ould cause unnecessary complications in the pro ofs. Lemma 3.2. L et G b e a c onne cte d planar gr aph, C ( G ) the c on-tr e e of G , and v 1 , v 2 , w 1 , w 2 ∈ V ( G ) . Consider any optimal emb e dding pr efer enc es Π and Π 0 of { v 1 , v 2 } and { w 1 , w 2 } , r esp e ctively. Construct alternative emb e dding pr efer enc es Π 00 of { w 1 , w 2 } as fol lows: use the emb e dding pr efer enc es of Π for al l no des at which P ( Q v 1 ,v 2 ) , P ( Q w 1 ,w 2 ) ar e c oher ent, and those of Π 0 for al l other no des of P ( Q w 1 ,w 2 ) . Then, Π 00 is again optimal for { w 1 , w 2 } . Pr o of. Let P = P ( Q v 1 ,v 2 ) and P 0 = P ( Q w 1 ,w 2 ) b e the considered con-paths. W e iterativ ely change Π 0 to Π 00 . W e consider the no des ν ∈ V ( P ) ∩ V ( P 0 ) at which P , P 0 are coherent one by one, and claim by induction that at each step (the order of which is unimp ortan t) the current em b edding preference Π 00 is optimal for { w 1 , w 2 } . The claim is trivial if ν holds the void preference anyho w. Let µ 1 , µ 2 b e the t wo neighbors of ν on R . Assume that ν is a P-no de. By Deﬁnition 2.15, the em b edding preferences of Π and Π 0 at ν are the same, since µ 1 , µ 2 are the same neighbors of ν on b oth paths P , P 0 . Hence, nothing changes at this step. By the same argument the preference at ν remains unchanged when ν is a C- or S-no de, unless one or b oth of µ 1 , µ 2 is an R-no de. It remains to consider the case that ν is a C- or S-no de and µ i is an R- no de for some i ∈ { 1 , 2 } . Let σ i b e the neighbor of µ i other than ν on R (since µ i is not an end by Deﬁnition 3.1). Let a i b e the element of the skeleton S µ i corresp onding to ν and b i the element of S µ i corresp onding to σ i ( a i is a cut v ertex if ν is a C-no de, but a virtual edge if ν is an S-no de; analogously for σ i ). An optimal solution to the single edge insertion of { v 1 , v 2 } requires to ﬁnd a shortest weigh ted dual path lo cally in the rigid skeleton S µ i from some face ϕ i inciden t with a i to a face ψ i inciden t with b i . This lo cal setting is the same for P 0 as for P , and so the same dual path from ϕ i to ψ i within S µ i can b e taken also in an optimal solution to the insertion of { w 1 , w 2 } . So, we get that the embedding preference of Π 0 at a C-no de ν (which lists some face of S µ i inciden t with cut vertex a i ) can b e c hanged in Π 00 to that of Π at ν (whic h lists ϕ i as the face incident with a i ) while preserving optimality . This is simultaneously done for i = 1 , 2 if b oth µ 1 , µ 2 are R-no des. Similarly for an S-node ν ; a choice of one of the t wo faces of S µ i inciden t with virtual edge a i determines the spin of a i , whic h consequently sho ws that the switc hing attribute of Π 0 at ν can b e changed to that of Π at ν while preserving optimality . u t 19 R emark 3.3. Algorithmically , it is trivial to ensure that multiple calls to the same dual shortest-path subproblem { a, b } within an R-no de skeleton alwa ys giv e the same solution. This can, e.g., b e achiev ed picking the start vertex for the searc h from { a, b } in a deterministic fashion (e.g., based on some arbitrary indexing), and using a deterministic algorithm for the dual path search. Using this metho d and with resp ect to Lemma 3.2, one can easily ensure that all the optimal em b edding preferences computed individually by Theorem 2.19 already agree at an y coherent no des. Since we deal with m ultiple edge insertion in our pap er, we hav e to handle situations when not all the optimal embedding preferences of the pairs from F can simultaneously b e satisﬁed in any plane embedding of G . W e use the follo wing terminology . Deﬁnition 3.4. Let Π U b e em b edding preferences of any set U of con-tree no des. W e sa y that an embedding G 1 honors the pr efer enc es Π U with defe ct r if there exists a subset U 0 ⊆ U of size | U 0 | = | U | − r such that G 1 honors (Deﬁnition 2.16) the restriction Π U 0 . Lemma 3.5. L et G b e a c onne cte d planar gr aph, v 1 , v 2 ∈ V ( G ) , and Π v 1 v 2 optimal emb e dding pr efer enc es of { v 1 , v 2 } . Assume that G 1 is an emb e dding of G such that G 1 honors Π v 1 v 2 with defe ct r ≥ 0 . Then it is p ossible to dr aw a new e dge f = v 1 v 2 into G 1 with at most ins( G, f ) + r · b ∆ ( G ) / 2 c cr ossings. Before w e pro ve this lemma, w e ma y informally describe its statemen t as follo ws: F or every individual preference on the con-chain of { v 1 , v 2 } that is not honored b y G 1 in the sense of Deﬁnition 3.4, we can apply one “ r ep air op er ation ” to f costing at most b ∆ ( G ) / 2 c new crossings (o ver the optimal insertion). The follo wing is needed in the pro of: Claim 3.6. L et P v 1 ,v 2 b e the c on-p ath of a p air { v 1 , v 2 } . Consider an internal P-, C-, or S-no de σ ∈ V ( P v 1 ,v 2 ) , and let w b e a vertex of the skeleton S σ . (a) ins( G, v 1 w ) + ins( G, wv 2 ) ≤ ins( G, v 1 v 2 ) . (b) L et Π v 1 v 2 b e optimal emb e dding pr efer enc es of { v 1 , v 2 } and Π v 1 w b e their r estriction to the internal no des of P v 1 ,w ( P v 1 ,v 2 of the p air { v 1 , w } . Then, Π v 1 w is an optimal emb e dding pr efer enc e of { v 1 , w } . Pr o of. Part (a) follows from the fact that there is a vertex cut X ⊆ V ( S σ ) in G with w ∈ X , separating v 1 from v 2 : if σ is a C-no de (of the cut vertex w ) then X = { w } ; otherwise, there is a suitable 2-cut X = { w , x } in the skeleton S σ —the latter is a cycle or a parallel bunch. Hence, in an y plane embedding G 0 of G , the edge f = v 1 v 2 has to b e drawn through a face ϕ incident with w , and the edges f 1 = v 1 w , f 2 = w v 2 can be drawn along the arc of f up to this face ϕ , giving an upp er b ound on ins( G, f 1 ) + ins( G, f 2 ) ≤ ins( G 0 , f ). Sp ecially , for G 0 that minimizes ins( G 0 , f ) we get ins( G, f 1 ) + ins( G, f 2 ) ≤ ins( G 0 , f ) = ins( G, f ). W e prov e (b) by means of contradiction. Assume f is drawn optimally (with ins( G, f ) crossings) into a suitable plane embedding G 0 of G , and let f 1 and f 2 b e dra wn in G 0 as in the previous paragraph with c 1 and c 2 crossings eac h. Then 20 c 1 + c 2 = ins( G, f ) due to (a) and optimalit y . If Π v 1 v 2 restricted to P v 1 ,w w as not optimal, then ins( G, f 1 ) < c 1 w ould b e ac hieved by some embedding G 1 of G . But then, there is a suitable edge order and/or ﬂipping decision at σ to com bine G 0 and G 1 across the cut X (from the previous paragraph) such that inserting f requires no crossings at S σ and consequently only ins( G, f 1 ) + c 2 < ins( G, f ) crossings altogether—a con tradiction. u t Pr o of (of L emma 3.5). Let P v 1 ,v 2 b e the con-path (in the decomp osition graph of G ) of the pair { v 1 , v 2 } . W e prov e the lemma by induction on r . The base case r = 0 is already established by Theorem 2.19. W e choose the ﬁrst no de σ on P v 1 ,v 2 from the v 1 -end such that Π v 1 v 2 at σ is not honored by G 1 . Then σ is an internal P-, C-, or S-no de of P v 1 ,v 2 . Let w ∈ V ( G ) b e an y v ertex of the sk eleton S σ , and P v 1 ,w , P w,v 2 b e the con-paths of { v 1 , w } and { w , v 2 } , resp ectiv ely . Note that P v 1 ,w , P w,v 2 ⊆ P v 1 ,v 2 , and that—by Claim 3.6(b) and symmetry—the corresp onding restrictions Π v 1 w , Π wv 2 arising from Π v 1 v 2 are optimal embedding preferences of { v 1 , w } and { w , v 2 } , resp ec- tiv ely . Consequen tly , G 1 honors Π v 1 w as whole (i.e. with defect 0) and Π wv 2 with defect r − 1. So f 1 = v 1 w can b e drawn into G 1 with ins( G, f 1 ) crossings by Claim 3.6(b) and f 2 = w v 2 can b e dra wn into G 1 with at most ins( G, f 2 ) + ( r − 1) · b ∆ ( G ) / 2 c crossings b y the induction assumption. Altogether, using Claim 3.6(a), G 1 + f 1 + f 2 is drawn with at most ins( G, f ) + ( r − 1) · b ∆ ( G ) / 2 c crossings, each one o ccuring on e ither f 1 or f 2 . Consider the arc g arising from joining the edge arcs representing f 1 and f 2 . W e would like to use g to draw f but g passes through the vertex w (where f 1 , f 2 meet). By a tin y p erturbation of g in a small neigh b orho od of w in G 1 w e can obtain a prop er drawing of G 1 + f at a cost of at most b ∆ ( G ) / 2 c additional crossings with (at most half of the) edges inciden t to w . This establishes ins( G 1 , f ) ≤ ins( G, f ) + r · b ∆ ( G ) / 2 c u t 3.2 The approximation algorithm Algorithm 3.7 (Solving MEI with additive appro ximation guarantee). Consider an instance of the multiple edge insertion problem MEI( G, F ): given is a connected planar graph G and a set F of k pairs of vertices of G suc h that F ∩ E ( G ) = ∅ . (1) Build the con-tree C := C ( G ). (2) Let F =  { u i , v i } : i = 1 , 2 , . . . , k  . F or i = 1 , 2 , . . . , k , determine the ( unique ) con-chain of { u i , v i } in C and the corresp onding con-path P i with the ends α i , β i , and, indep endently for eac h i , call the algorithm of The- orem 2.19 to compute optimal em b edding preferences Π i of { u i , v i } (see Remark 3.8 for consistency). (3) Denote by p ( ν ) := { i : ν ∈ V ( P i ) \ { α i , β i }} the set of indices of all the pairs from F that ha ve a preference (p ossibly void) at a con-tree no de ν . F or eac h ν of C , choose (suitably) a subset p 0 ( ν ) ⊆ p ( ν ) according to some rules deﬁned later on, see Remark 3.9 for details. 21 (4) F or i ∈ { 1 , 2 , . . . , k } , let π i ( ν ) denote the individual preference (if existent) of Π i at ν . Let R ( ν ) := { π i ( ν ) : i ∈ p 0 ( ν ) } b e the multiset of the individual preferences at ν requested b y (only) those relev ant con-paths that hav e b een selected in step (3). a) F or each no de ν of C , if R ( ν ) = ∅ then set a resulting preference π ν arbitrarily or v oid. Otherwise, choose a preference π ν ∈ R ( ν ) such that π ν is among the elements with maximum multiplicit y in R ( ν ) (a semi- majority choic e ). b) Using Lemma 3.14 (see b elo w), compute a plane embedding G 0 of G that honors the em b edding preferences Π := { π ν : ν ∈ V ( C ) } . (5) Indep enden tly for each i = 1 , 2 , . . . , k , compute (deterministically; see Re- mark 3.11) the insertion path for { u i , v i } in to the ﬁxed embedding G 0 . W e defer a detailed runtime analysis of this algorithm to the end of the section. First, w e would lik e to comment on some of the steps in this algorithm: R emark 3.8 (Step (2) of Algorithm 3.7). Based on Remark 3.3, we can assume that our algorithm pro duces consisten t embedding preferences at coheren t nodes (for any pair of con-paths) without further treatment. If, for an y reason, one does not wan t to honor Remark 3.3, there is a simple algorithmic work around. F or i = 2 , 3 , . . . , k ; for any j < i and any no de ν of C such that P i , P j are coherent at ν , change the individual preference of Π j at ν to that of Π i at ν . By Lemma 3.2, the ﬁnal so-mo diﬁed embedding preferences (still denoted by Π j ) are still optimal for their resp ectiv e pairs { u j , v j } . R emark 3.9 (Step (3) of Algorithm 3.7). The practical meaning of step (3) is that we may choose to “ignore” some (or even all) of the individual preferences requested by (some of ) the con-paths. Herein, w e will closely discuss tw o v ariants. W e may choose p 0 ( ν ) ⊆ p ( ν ) – arbitr arily , as long as p 0 ( ν ) 6 = ∅ if p ( ν ) 6 = ∅ ; or – sp eciﬁcally , to fulﬁll up coming Deﬁnition 3.18. Although in particular the ﬁrst v ariant may sound silly , there are actually tw o go od reasons for allowing freedom in the choice of p 0 ( ν ). First, this leav es plen ty of ro om for (heuristic) algorithm engineering in practical applications, while still pro viding a ﬁrm approximation guarantee for essentially any somewhat reason- able choice of p 0 ( ν ) (the ﬁrst v ariant, Prop osition 3.13). Observe that the ﬁrst v ariant also co vers the p ossibilit y of simply setting p 0 ( ν ) := p ( ν ) for all ν ∈ V ( C ), and also the p erhaps most na ¨ ıv e choice, picking one arbitrary i ∈ p ( ν ) and set- ting p 0 ( ν ) := { i } . Second, for a carefully crafted (and still eﬃcient) choice of p 0 ( ν ) w e can in fact provide a stronger worst-case guaran tee (the second v ariant, Theorem 3.19) than if w e considered all the preferences together. R emark 3.10 (Step (4)a) of A lgorithm 3.7). W e do not perform any further optimization of the choice of π ν in the algorithm here, even though it can b e p ossible that some embedding speciﬁcation could b e simultaneously go o d for sev eral distinct individual preferences at ν (again, this lea ves ro om for further 22 p ossibly heuristic algorithm engineering). The presented semi-ma jority choice is just righ t to prov e the algorithm’s ov erall approximation ratio. R emark 3.11 (Step (5) of Algorithm 3.7). By using a deterministic shortest path algorithm in the dual of G 0 , we can trivially ensure that distinct insertion paths do not cross m ultiple times. If, for some reason, we do not apply a suﬃcien tly deterministic algorithm, we can simply exchange subpaths as a p ostpro cessing step, suc h that in the end all inserted edges cross each other at most once. While deferring the lengthy implemen tation details of step (4) to Lemma 3.14, w e illustrate the underlying idea of Algorithm 3.7 with the following simple claim and its corollary in Prop osition 3.13. Claim 3.12. Consider the setting of Algorithm 3.7. F or any i 6 = j ∈ { 1 , . . . , k } , ther e ar e at most two no des ν of C ( G ) such that b oth π i ( ν ) , π j ( ν ) exist and π i ( ν ) 6 = π j ( ν ) , i.e., the c ompute d optimal pr efer enc es Π i and Π j r e quest diﬀer ent individual pr efer enc es at ν . Pr o of. Since individual em b edding preferences are stored at the corresp onding con-path no des, conﬂicting preferences may only arise on R := P i ∩ P j , whic h is a path by Claim 2.11. W e kno w that Π i , Π j agree at all coherent no des in R (either due to the deterministic algorithm or after applying Lemma 3.2). By Deﬁnition 3.1 of coherence, p ossible conﬂicts could only b e at (a) the ends of R and (b) no des neigh b oring an R-no de that is an end of R . Recall that R-no des store the void embedding preference, and so at each end of R there can only b e one troublesome no de, either of type (a) or (b). This establishes the claim. u t Prop osition 3.13 (W eak estimate). Consider a c onne cte d planar gr aph G and a set F of k vertex p airs over V ( G ) . L et ins Σ ( G, F ) := P f ∈ F ins( G, f ) b e the sum of the individual insertion values—an obvious lower b ound for ins( G, F ) . If Algorithm 3.7, in step (3), cho oses arbitrary p 0 ( ν ) 6 = ∅ for e ach no de ν of C ( G ) with p ( ν ) 6 = ∅ , then the r esult is a plane emb e dding G 0 of G such that ins( G, F ) ≤ ins( G 0 , F ) ≤ ins Σ ( G, F ) +  2  ∆ ( G ) 2  + 1  ·  k 2  . (6) Note that already this short statemen t establishes the approximation factor given for the MEI problem in the conference version of this pap er [8]; herein (Theo- rem 3.19) w e will later establish a stronger b ound as well. Pr o of. W e wan t to sho w, in the language of Deﬁnition 3.4, that the sum of defects of G 0 honoring each one of the optimal preferences Π i , i = 1 , . . . , k , from the algorithm is at most 2  k 2  . Inequalit y (6) w ould then immediately follo w from Lemma 3.5 and the fact that the edges of F pairwise cross at most once. Assume the notation of Algorithm 3.7. Let ˆ G 0 b e a go o d enric hed em b edding for Π . W e say a pair ( µ, i ), for an y µ ∈ V ( C ) and 1 ≤ i ≤ k , forms a dirty p ass if ˆ G 0 is not go od for π i ( µ ) (Deﬁnition 2.16). Obviously , the total num b er of dirty passes equals the sum of defects of G 0 . W e show that all the dirty passes of ˆ G 0 23 can b e counted tow ards unordered index pairs { i, j } ⊆ { 1 , . . . , k } such that each suc h pair is “responsible” for at most t w o dirty passes altogether. Indeed, if ( ν, i ) is a dirty pass, then there exists some j ∈ { 1 , . . . , k } suc h that the computed preference at ν is π ν = π j ( ν ) 6 = π i ( ν ). W e hence coun t ( ν , i ) tow ards { i, j } and, b y Claim 3.12, we already know that this ma y happ en at most twice for eac h pair { i, j } . u t Finally , we provide a straightforw ard implementation and a pro of of correct- ness of step (4)b) in Algorithm 3.7. Lemma 3.14. L et G b e a c onne cte d planar gr aph and C = C ( G ) a c on-tr e e of G . Assume Π = { π ν : ν ∈ V ( C ) } is an arbitr ary c ol le ction of no de emb e dding pr efer enc es for C . Then ther e is a plane emb e dding G 0 of G such that G 0 honors the pr efer enc e Π . The emb e dding G 0 c an b e c ompute d in line ar time. Pr o of. In the ﬁrst step we ﬁx a plane embedding H 1 of each blo c k H of G . If H is trivial, its embedding is already pre-sp eciﬁed (or even unique). Otherwise H 1 is determined b y deducing an embedding together with all the spin v alues. Let T 1 ⊆ T ( H ) b e a subtree of the sSPR-tree of H stored in C . W e prov e by induction on | V ( T 1 ) | that there exists a go o d enriched embedding H 1 of H for Π restricted to (the no des of ) T 1 . The claim holds for empty T 1 . Let ν b e a leaf of T 1 , and let H 2 b e a goo d enric hed embedding of H for Π restricted to T 1 − ν . Consider the t yp e of ν as in Deﬁnition 2.4: a) If ν is an S-no de, then H 1 := H 2 and we only determine the asso ciated spin v alues. Let { µ 1 , µ 2 } b e the π ν -p eers stored at ν , and e µ 1 , e µ 2 the resp ectiv e virtual edges in S ν . Actually , if µ i , i ∈ { 1 , 2 } , is a C-no de, w e simply refer as e µ i to the corresp onding CS-pair. Up to symmetry , µ 2 6∈ V ( T 1 ). If µ 1 6∈ V ( T 1 ) as well, we set the spin of e µ 1 arbitrarily (while otherwise it has already b een set by µ 1 ). In any case, we select the spin of e µ 2 to honor the switching / nonswitching lab el at π ν , and we charge the remaining spins asso ciated with ν arbitrarily . b) If ν is an R-no de, it may b e that its neighbor µ in T 1 is an S-no de such that ν is one of the π µ -p eers of µ . If this is the case, then we get H 1 b y ﬂipping the embedding of S ν in H 2 suc h that the spin v alue and switching lab el sp eciﬁed by µ are honored. Otherwise, let H 1 := H 2 . W e also set the spin v alues of the virtual edges in S ν accordingly . c) If ν is a P-node, then the π ν -p eers are S-no des { µ 1 , µ 2 } where, up to sym- metry , µ 2 6∈ V ( T 1 ). W e arbitrarily set the spin v alues of all the virtual edges e 6 = e µ 2 of S ν , except p ossibly e = e µ 1 if µ 1 ∈ V ( T 1 ) (then the spin has b een set by µ 1 ). F or H 1 w e choose an embedding of S ν , and a spin v alue of e µ 2 , suc h that e µ 1 , e µ 2 form a face as required p er Deﬁnition 2.16, case (P1). No w, we hav e an enriched embedding G 1 of G with the prop ert y that, for eac h blo ck H of G , G 1 induces an enriched subembedding H 1 that is go o d for Π restricted to T ( H ). T o obtain the ﬁnal embedding G 0 of G , it remains to mo dify G 1 —only at the cut vertices of G —suc h that resulting G 0 is go o d for Π at all 24 the C-no des of C . Note that, technically working with spherical embeddings, we can freely choose the outer face of any H 1 ⊆ G 1 in the plane without changing the em b edding sp eciﬁcation at any no de of T ( H ). W e again pro ceed by induction on the size of a suitable subtree B 2 ⊆ B ( G ) suc h that there exists an embedding G 2 of G go od for Π restricted to all the C-no des and all the sSPR-trees of B 2 . F or tec hnical reasons, B 2 needs to b e sp e cial , meaning that all the leav es of B 2 are B-no des (which clearly holds true for whole B ( G )). The base of the induction is B 2 formed by a singleton B-no de of a blo c k H , for which the sub em b edding H 0 ⊆ G 2 := G 1 has b een ﬁxed abov e. In the induction, any arising B 2 will contain a C-node γ , suc h that if B 3 ⊆ B 2 results by removing γ and all its adjacen t B-leav es, then B 3 is empty or again a sp ecial tree. By the induction assumption, let G 3 b e an embedding of G that is go od for Π restricted to B 3 . Let c ∈ V ( G ) b e the cut vertex represented by γ , and { µ 1 , µ 2 } b e the π γ -p eers at γ where µ i b elongs to the adjacent sSPR-tree T ( H i ), i = 1 , 2. F or i = 1 , 2, w e take a sp eciﬁc face ϕ i of the skeleton S µ i : for a D-no de µ i , w e take its default face; for an R-no de, ϕ i is sp eciﬁed by the lab el in π γ ; for an S-no de, ϕ i is determined by the spin v alue of the CS-pair ( c, µ i ). Let ϕ 0 i b e the corresp onding face of the (sub)embedded blo c k H i ⊆ G 3 . W e may assume, up to symmetry , that µ 2 do es not belong to an sSPR-tree held by B 3 . W e mak e ϕ 0 2 the outer face of H 2 , and we construct G 2 from G 3 b y rearranging the embedding sp eciﬁcation at γ (and, thereby , the blo c ks incident with c ) such that H 2 o ccurs inside the face ϕ 0 1 . It is clear that G 2 is no w go o d also for π γ . Finally , follo wing the constructive steps of this pro of, it is routine to verify that the resulting em b edding G 0 can b e computed in ov erall linear time. u t 3.3 Impro ved appro ximation guarantee W e are going to turn the weak approximation guarantee of Prop osition 3.13 in to an asymptotically optimal one—with the additiv e O ( ∆ · k 2 ) term improv ed do wn to O ( ∆ · k log k + k 2 ). T o achiev e this goal, we will count the dirty passes of G 0 , and so the ov erall defect, similarly to the pro of of Prop osition 3.13, but with resp ect to a special order of the con-paths such that, at e ac h step, we accoun t for roughly at most O (log k ) new ones. The tw o crucial ingredients for this approac h are the semi-ma jorit y c hoice of the preferences Π in Algorithm 3.7 and the follo wing folklore 7 claim. Claim 3.15. L et T b e a tr e e, and U i ⊆ T , i = 1 , 2 , . . . , h , b e an arbitr ary c ol le ction of subtr e es of T . Then ther e exists j ∈ { 1 , . . . , h } and u ∈ V ( U j ) such that, for every i ∈ { 1 , . . . , h } , if U i interse cts U j then u ∈ V ( U i ) . Pr o of. Consider any graph H , subgraph H 0 ⊆ H , and v ∈ V ( H ). Let the distanc e fr om v to H 0 b e the minimum distance d b et ween v and a vertex of H 0 , and the shor e of H 0 fr om v b e the subset of those vertices of H 0 ha ving distance d from v . 7 This claim is b etter known in the following formulation: The intersection graph of subtrees in a tree is chordal and it contains a so-called simplicial vertex. 25 Clearly , if H is a tree and H 0 is connected, then the shore of H 0 m ust alwa ys b e a single v ertex u and every path from v to H 0 con tains u . Cho ose any v ertex v ∈ V ( T ) and let d 0 b e the maximum distance from v to any U i , i ∈ { 1 , . . . , h } . If d 0 = 0 then u := v fulﬁlls the claim for any j . Supp ose d 0 > 0 and let some U j , j ∈ { 1 , . . . , h } , with the shore u ∈ V ( U j ), be at distance d 0 from v . W e show that this c hoice of u, j fulﬁlls the claim. Consider an y U i , i ∈ { 1 , . . . , h } , that intersects U j . Clearly its distance from v is at most d 0 . Let x ∈ V ( U i ) ∩ V ( U j ) and let y b e the shore of U i ∪ U j from v . According to the previous paragraph, the path from v to x m ust contain b oth of y , u , and so u ∈ V ( U i ), to o. u t W e ﬁrst brieﬂy outline how Claim 3.15 can help to improv e the estimate for Algorithm 3.7 o ver Prop osition 3.13: a) F or an instance MEI( G, F ), assume that the decomp osition graph D := D ( G ) is a tree (this happ ens, e.g., when G is 2-connected). Let P i ⊆ D , i = 1 , . . . , k , b e the con-paths of the k insertion pairs { u i , v i } ∈ F . Then, b y Claim 3.15, there is j ∈ { 1 , . . . , k } such that all of the con-paths hitting P j do so in the same no de ν ∈ V ( P j ). b) Let P 0 j , P 00 j b e the t wo half-paths of P j from a) starting in ν , and let ` < k b e the total num ber of con-paths sharing ν with P j . W e can claim that the em- b edding G 0 computed in Algorithm 3.7 honors the preferences Π j restricted to P 0 j with defect at most log 2 ` < log 2 k : informally , by our semi-ma jority c hoice, the embedding might not b e go o d only for individual preferences of Π j at those no des of P 0 j where at least half of all participating con-paths “div ert from” P 0 j (at this no de or at the next R-no de). An upp er b ound log 2 k for each of P 0 j , P 00 j hence follo ws easily , and we may account for +1 in the defect sum due to ν itself. c) If the previous b ound was extendable recursiv ely to all the con-paths, a ﬁnal estimate for the sum of all the defects would b e at most 1 + 2 log 2 k + 1 + 2 log 2 ( k − 1) + · · · ≤ k + 2 k log 2 k (while the actual b ound in Theorem 3.19 will come out just sligh tly worse). F or an informal reference, w e will call this approach a log 2 k -defe ct ar gument . There are, ho wev er, tw o big problems with the outlined (optimistic) scenario: R emark 3.16 (On utilizing the log 2 k -defe ct ar gument). a) Since D ( G ) do es not hav e to b e a tree, Claim 3.15 do es not directly hold for it (ho wev er, we at least know that tw o con-paths may only intersect in a subpath in D ( G )). b) A somehow less obvious problem with the log 2 k -defect argumen t p ops up when applying it recursively—after removing sev eral of the con-paths, it is no longer true that a semi-ma jorit y c hoice in this sub collection is the same as the semi-ma jority choice made by the algorithm (for all the con-paths). T o address Remark 3.16 a), we reﬁne Claim 3.15 in follo wing Lemma 3.17. Note, how ev er, that if the given graph G would b e biconnected, D ( G ) would b e 26 S S S S P S C R S γ R R R (a) W e could reroute the dotted con-path to b ecome the dashed con-path, without problems. W e require no embedding pref- erences at the newly added nodes, as their sk eletons all share the cut vertex repre- sen ted by γ . S S S S P S C R S γ R R R (b) F or each of the three con-paths, we mark those of their no des with a double circle for whic h γ is a substitute. S S S S P S C R S γ R R P i P j μ R (c) In a situation like this, Claim 3.15 w ould fail, as the graph induced by the con-paths is not a tree. Lemma 3.17 allo ws to pick P j and µ (double circle): for the la- b eled P i w e hav e V ( P i ) ∩ V ( P j ) 6 = ∅ but it do es not contain µ ; how ever, γ ∈ V ( P i ) is a substitute for µ w.r.t. P j . Fig. 2. Substitutes w.r.t. con-paths. W e visualize a part of a decomp osition graph D , analogously to Figure 1. Con-paths are shown as thick dotted/dashed curves. The mates of the central C-no de γ are marked with thic k b orders. a tree and w e could directly use Claim 3.15 and its consequen t b ound instead. Consider a con-path P and an internal no de ν ∈ V ( P ). Recall the notion of a mate from Deﬁnition 2.7, and the fact that a mate of a cut vertex x ma y also b e a P-no de in which case it is not directly adjacen t to x ’s C-no de in D . W e say that a no de γ ∈ V ( D ) is a substitute for ν w.r.t. P if (cf. Figure 2): – γ is a C-no de of a cut vertex x of G such that ν is a mate of x (and so ν is not a C-no de), and – there is a neigh b or µ of ν on P such that µ = γ or µ is a mate of x , to o. In the following lemma, we consider some index set I . While it is most natural to think of I as { 1 , . . . , k } , w e will later apply this lemma also for subsets of the latter. Lemma 3.17. L et G b e a c onne cte d planar gr aph, I b e a set of indic es, and P i , i ∈ I , b e a c ol le ction of c on-p aths in the de c omp osition gr aph D := D ( G ) (as 27 derive d fr om an arbitr ary c ol le ction of c on-chains in C ( G ) ). Then ther e exists j ∈ I and µ ∈ V ( P j ) such that, for al l i ∈ I , the fol lowing holds: if V ( P i ) ∩ V ( P j ) 6 = ∅ , then µ ∈ V ( P i ) or V ( P i ) c ontains a substitute for µ w.r.t. P j . Note that diﬀerent P i ’s in the statemen t ma y hav e diﬀerent substitutes for µ , but there are at most four av ailable substitutes for µ w.r.t. P j an ywa y (one for eac h vertex of the tw o virtual edges of the neighbors of µ on P j ). Pr o of. W e would lik e to apply Claim 3.15 but D is not a tree (due to adjacencies of the C-nodes, see Deﬁnition 2.7 and Figure 2(c)). Consider a C-no de γ ∈ V ( D ) of a cut v ertex x of G . If H ⊆ G is a blo c k of G inciden t with x , then the set of all mates of x in T ( H ) induces a subtree T γ ( H ) ⊆ T ( H ). W e construct a spanning tree D 0 ⊆ D as follo ws: – F or every C-no de γ of D and each incident blo c k H ⊆ G , we delete all but (an arbitrary) one of the edges of D b et w een γ and V ( T γ ( H )). – F or eac h P i , i ∈ I , such that P i passes through any such γ and one of its deleted edges, we reroute P i as P 0 i along the remaining edge (b et ween γ and T γ ( H )) and through T γ ( H ). Otherwise, let P 0 i := P i . No w we apply Claim 3.15 onto D 0 and the paths P 0 i , i ∈ I , obtaining a pair j and µ 0 ∈ V ( P 0 j ). W e analyze the situation with resp ect to D and P j . – If µ 0 ∈ V ( P j ), then we set µ := µ 0 . Otherwise, µ 0 is not a C-no de and there is a C-no de γ ∈ V ( P j ) such that µ 0 is a mate of the cut v ertex of γ ; then, we set µ to b e the neighbor of γ on P j corresp onding to µ 0 (i.e., µ is the ﬁrst no de in P j when tra versing P 0 j starting at µ 0 and a wa y from γ ). – If, for any i ∈ I , µ 0 6∈ V ( P 0 i ) then V ( P i ) ∩ V ( P j ) = ∅ ; this is since V ( P i ) ⊆ V ( P 0 i ) by our construction. (Though, the conv erse direction is not true in general.) Consequently , our analysis only has to consider those indices i ∈ I for whic h µ 0 ∈ V ( P 0 i ) and V ( P i ) ∩ V ( P j ) 6 = ∅ . – If µ ∈ V ( P i ) then we are done. This is true, in particular, when µ is a C- no de. Hence, in addition to the previous paragraph, w e may now assume that µ is not a C-no de and µ 6∈ V ( P i ). Let H b e the block of G such that µ ∈ T ( H ). – If µ = µ 0 then, since µ ∈ V ( P 0 i ) \ V ( P i ), the path P 0 i has b een rerouted from original P i at a C-no de γ 0 ∈ V ( P i ) such that µ = µ 0 ∈ V ( T γ 0 ( H )). Let τ ∈ T ( H ) b e the neighbor of γ 0 on P i ( τ is the only no de of P i in T γ 0 ( H )). Since P i in tersects P j and T ( H ) has no cycles, it is τ ∈ V ( P j ) \ { µ } . Consequen tly , γ 0 is a substitute for µ w.r.t. P j b y deﬁnition. – It remains to consider, in addition to the previous, that µ 6 = µ 0 . In this case the C-no de γ has b een deﬁned ab o ve; we ha ve γ µ ∈ E ( P j ) and γ is a substitute for µ w.r.t. P j . So, if γ ∈ V ( P i ) then we are done. Hence µ, γ 6∈ V ( P i ), and (a) P i in tersects P j in T ( H ) in a no de % 6 = µ , or (b) P i is disjoin t from P j in T ( H ) but they intersect in a C-no de δ 6 = γ inciden t with the blo c k H . Consider case (a); since the only connection from µ 0 to % in T ( H ) contains µ (and µ 6∈ V ( P i )), there again has to b e a C-no de γ 0 ∈ V ( P i ) such that 28 µ 0 , % ∈ V ( T γ 0 ( H )), and consequently µ ∈ V ( T γ 0 ( H )) and γ 0 is a substitute for µ w.r.t. P j , to o. In case (b), P 0 i ∩ P 0 j con tains a path from µ 0 to δ (since δ has only one edge to T ( H ) in D 0 ), while the P i -neigh b or of δ in T ( H ) b elongs to V ( P 0 i ) \ V ( P j ). Therefore, µ ∈ V ( T δ ( H )) and so µδ ∈ E ( P j ) by minimalit y . Then δ is a substitute for µ w.r.t. P j . u t An application of Lemma 3.17 in the aforementioned log 2 k -defect argument brings another elusiv e problem; namely , what should b e the subpaths P 0 j , P 00 j of selected P j to which the argumen t is applied? The crucial prop ert y w e need is that ev ery con-path P i hitting P 0 j do es so in the starting no de of P 0 j . Informally , for P j and µ as in Lemma 3.17 w e let P o j ⊆ P j b e the minimal subpath intersecting all the con-paths not disjoint from P j . Then P 0 j , P 00 j can b e chosen as the subpaths edge-disjoin t from P o j suc h that P 0 j ∪ P o j ∪ P 00 j = P j . Ho wev er, we also hav e to consider p ossible defects (an unbounded num b er of ?) due to honoring the preferences of P o j . Let α o , β o b e the ends of P o j . W e will sho w that each of the corresp onding skeletons S α o , S β o shares a vertex with S µ (formal details giv en later). Consequently , only at most tw o repair op erations (at these shared vertices) are enough for the whole insertion path along P o j , while the individual preferences along P o j are simply ignored in the algorithm (Remark 3.9). This accoun ts for at most 2 log 2 k + 2 dirty passes along whole P j , as desired. Finally , addressing Remark 3.16 b) is actually not that diﬃcult, and we in- formally outline the idea now. Cho ose any node ν of D ( G ) and fo cus exclusively on ν and the con-paths passing through it, in other words, on the multiset R ( ν ) = { π i ( ν ) : i ∈ p 0 ( ν ) } of (selected) individual preferences at ν from Algo- rithm 3.7. Let π ν ∈ R ( ν ) b e the individual preference chosen in the algorithm. Without loss of generalit y we ma y assume that the rep eated application of Lemma 3.17 selects and remov es con-paths in the order P 1 , P 2 , . . . , P k . W e deﬁne R 1 ( ν ) = R ( ν ) and, for i ∈ { 2 , 3 , . . . , k } , R i ( ν ) = R i − 1 ( ν ) \ { π i − 1 ( ν ) } (with mul- tiplicit y). The in tuitiv e problem with a na ¨ ıv e v oting sc heme (that would consider the full sets p ( ν ) instead of p 0 ( ν )) is that it would b e p erformed ov er R ( ν ) at eac h no de ν ; our rep eatedly applied counting argument based on Lem ma 3.17, ho wev er, requires a voting according to the corresp onding R i ( ν ) at each step. This can b e seen as follows: If π 1 ( ν ) 6 = π ν then the multiplicit y of π 1 ( ν ) in R ( ν ) is ≤ 1 2 | R ( ν ) | and, informally , we can aﬀord to pay for one repair ope ra- tion at ν within the log 2 k -defect argument. Otherwise ( π 1 ( ν ) = π ν ), no repair op eration is needed. F or i ∈ { 2 , 3 , . . . , k } , when π i ( ν ) = π ν or the multiplicit y of π i ( ν ) in R i ( ν ) is ≤ 1 2 | R i ( ν ) | , the argument is ﬁne again. Consider the contrary , that π i ( ν ) 6 = π ν and so the multiplicit y of π i ( ν ) in R ( ν ) is ≤ 1 2 | R ( ν ) | , while the multiplicit y of π i ( ν ) in R i ( ν ) is > 1 2 | R i ( ν ) | . This may happ en, though, it is an easy exercise to sho w that ev ery such i can b e accoun ted tow ards some j < i such that π j ( ν ) = π ν and the multiplicit y of π j ( ν ) in R j ( ν ) is ≤ 1 2 | R j ( ν ) | . In the latter case of j , the log 2 k -defect argument reserved one repair op eration at ν for P j but it w as not required there. In the formal pro of b elo w, we will say that P j issues a “repair tic ket” at ν whic h is subsequently used by P i for an actual repair op eration. 29 W e are ﬁnally ready to prov e our stronger b ound. First, we deﬁne what implemen tation of step (3) of Algorithm 3.7 is needed to get our stronger b ound. Deﬁnition 3.18 (Ignori ng some individual preferences based on a go o d simplicial sequence). Consider the decomp osition graph D of a connected planar graph G , and con-paths P 1 , . . . , P k in D . i) F or any p erm utation σ of the indices, we sa y that a sequence  ( P σ (1) , µ σ (1) ) , . . . , ( P σ ( k ) , µ σ ( k ) )  , where µ j ∈ V ( P j ), is a go o d simplicial se quenc e if the follo wing holds true for i = 1 , . . . , k : – for I = { σ ( i ) , σ ( i + 1) , . . . , σ ( k ) } , the choice j := σ ( i ) and µ := µ σ ( i ) satisﬁes the conclusion of Lemma 3.17. ii) Let p ( ν ) b e deﬁned as in Algorithm 3.7. W e say that a system of sets p 0 ( ν ) ⊆ p ( ν ), ν ∈ V ( D ), ignor es the se quenc e  ( P σ (1) , µ σ (1) ) , . . . , ( P σ ( k ) , µ σ ( k ) )  if the follo wing holds true for i = 1 , . . . , k and all ν of D : – if i ∈ p ( ν ), then i 6∈ p 0 ( ν ) if and only if one of the following holds true; ν = µ i , or ν is a C-no de neighboring µ i on P i , or there is a cut vertex x of G suc h that b oth ν and µ i are mates of x .  Note that a goo d simplicial sequence alw ays exists by Lemma 3.17, but it ma y not b e unique (which is not a problem for our arguments). Also observe that our p 0 , as now deﬁned in Deﬁnition 3.18, “ignores” many more individual preferences than originally suggested in the aforemen tioned sk etch of the pro of idea. This allo ws for a simpler deﬁnition and do es not hurt the argumen t. Most importantly , Deﬁnition 3.18 ii) is inv ariant on the order σ of the simplicial sequence—w e only use information that a certain no de µ i is assigned to P i , i ∈ { 1 , . . . , k } when deﬁning p 0 . Theorem 3.19 (Strong estimate, Theorem 1.2(3)). Consider a c onne cte d planar gr aph G and a set F of k vertex p airs over V ( G ) . L et ins Σ ( G, F ) := P f ∈ F ins( G, f ) ≤ ins( G, F ) . If step (3) of Algorithm 3.7 is implemente d such that the c ompute d set system p 0 ( ν ) , ν ∈ V ( D ) , ignor es some go o d simplicial se quenc e over the c on-p aths P 1 , . . . , P k , then the algorithm outputs a plane em- b e dding G 0 of G such that ins( G, F ) ≤ ins( G 0 , F ) ≤ ins Σ ( G, F ) +  ∆ ( G ) 2  · 2 k b log 2 2 k c +  k 2  . (7) Pr o of. W e assume all the notation of Algorithm 3.7. W e may also assume, with- out loss of generalit y , that the go od simplicial sequence considered when deﬁning p 0 is  ( P 1 , µ 1 ) , . . . , ( P k , µ k )  . Recall that the actual order of pairs in this sequence is signiﬁcant only for the following analysis and it is not used in any wa y in Al- gorithm 3.7, when deﬁning p 0 . Denote b y q ` ( ν ) := p 0 ( ν ) ∩ { `, ` + 1 , . . . , k } . F or j = 1 , . . . , k in this order, we argue as follows. By Deﬁnition 3.18, j 6∈ p 0 ( µ j ). Let α j , β j ∈ V ( P j ) b e the ends of P j . Denote by α 0 j ∈ V ( P j ) \ { α j } the no de closest to α j on P j and suc h that i 6∈ p 0 ( α 0 j ), and b y β 0 j ∈ V ( P j ) \ { β j } the one closest to β j suc h that j 6∈ p 0 ( β 0 j ). W e observ e, by Deﬁnition 3.18, that the 30 sk eleton of α 0 j (and analogously β 0 j ) has a common vertex with the skeleton of µ j . Let P 0 j , P 00 j ⊆ P j denote the subpaths of P j from α 0 j to α j , and from β 0 j to β j . If α j = µ j or β j = µ j , then P 0 j or P 00 j is undeﬁned and hence the follo wing argumen t is simply skipp ed for it. Note that V ( P 0 j ) ∩ V ( P 00 j ) ⊆ { µ j } if b oth P 0 j , P 00 j are deﬁned. The subsequent argumen t will b e given only for P 0 j , with understanding that the same symmet- rically applies to P 00 j . By Lemma 3.17 and the deﬁnition of a substitute, if a con-path P i , i > j , in tersects P j , then P i con tains α 0 j or β 0 j , or P i in tersects P j − V ( P 0 j ∪ P 00 j ). Hence, b y Claim 2.11, if P i in tersects P 0 j then α 0 j ∈ V ( P i ), and so we hav e q j ( α 0 j ) ⊇ q j ( α 1 j ) ⊇ · · · ⊇ q j ( α a j ) where α 0 j , α 1 j , . . . , α a j = α j denote the no des of P 0 j (of length a ) in this order. F or ν ∈ V ( P i ) let ˜ q i ` ( ν ) ⊆ q ` ( ν ) denote the subset of those m ∈ q ` ( ν ) for which P m is coherent with P i at ν or i = m . W e say that no de ν is diver gent for P i on level ` if ν ∈ V ( P i ) is not an R-no de and | ˜ q i ` ( ν ) | ≤ 1 2 | q ` ( ν ) | . Our k ey observ ation is that the num b er of no des of P 0 j that are divergen t for P j on level j , is at most b log 2 | q j ( α 0 j ) |c ≤ b log 2 k c : whenever a no de α i j ∈ V ( P 0 j ) is divergen t w e hav e, b y the deﬁnition of coherence, | q j ( α i +1 j ) | ≤ 1 2 | q j ( α i j ) | if α i +1 j is not an R-no de, and | q j ( α i +2 j ) | ≤ 1 2 | q j ( α i j ) | if α i +1 j is an R-no de (recall that R- no des are not counted as divergen t). This upp er b ound of ≤ b log 2 k c divergen t no des along P 0 j will b e used to b ound the total defect of G 0 in honoring the computed optimal em b edding preferences of edges in F . Let ν ∈ V ( P 0 j ). If π j ( ν ) 6 = π ν then, by the deﬁnitions and the semi-ma jority c hoice in the algorithm, ν is divergen t for P j on level 1. Unfortunately , it might happ en that ν is divergen t for P j on level 1 but not divergen t for P j on level j , and w e cannot simply account for the cost of this defect at ν on the same level j . T o resolve suc h cases, we use an amortized analysis which “borrows” for the cost from smaller lev els. This is formalized as follows: I) If ν is divergen t for P j on level j , then we issue one r ep air ticket to ν (the total num b er of tick ets issued along P 0 j is thus at most b log 2 k c , as desired). I I) If π j ( ν ) 6 = π ν , then w e use one of the repair tick ets issued to ν tow ards a total defect in honoring the optimal preferences of P j restricted to P 0 j . I II) Let m b e such that π m ( ν ) = π ν and deﬁne r j := max  | ˜ q i j +1 ( ν ) | − | ˜ q m j +1 ( ν ) | : j < i ≤ k  ≥ 0 . After ﬁnishing steps (I) and (I I) on level j , the num ber of av ailable (i.e., issued and not y et used) repair tick ets for ν is at least r j . W e claim that this amortized analysis is sound, more precisely , that every time we w ould like to use a repair tick et in (I I), there is one av ailable. In a “simple” situation, step (I I) would simply use the tick et just issued in (I) on the same level (this happ ens, e.g., for j = 1). Assume that no tick et is issued on lev el j , which means that ν is not divergen t for P j , and so | ˜ q j j ( ν ) | > 1 2 | q j ( ν ) | . If π j ( ν ) 6 = π ν , then | ˜ q m j ( ν ) | < 1 2 | q j ( ν ) | and so r j − 1 ≥ | ˜ q j j ( ν ) | − | ˜ q m j ( ν ) | > 0. By 31 (I II), w e hav e got an av ailable repair tick et from one of the previous levels for use on lev el j . It is th us enough to prov e (II I) b y induction on j ≥ 0. F or the base case j = 0 (i.e., before the pro cess starts), w e ha ve r 0 = 0 b y the semi-ma jority choice of π ν . F urther on, let i > j b e suc h that r j = | ˜ q i j +1 ( ν ) | − | ˜ q m j +1 ( ν ) | . It suﬃces to discuss the inductiv e step in tw o cases, either r j − r j − 1 ≥ 1 or r j = r j − 1 > 0: – Assume r j − r j − 1 ≥ 1. Since r j − 1 ≥ | ˜ q i j ( ν ) | − | ˜ q m j ( ν ) | and ˜ q i j +1 ( ν ) ⊆ ˜ q i j ( ν ), w e ha ve | ˜ q m j +1 ( ν ) | = | ˜ q m j ( ν ) | − 1. Then π j ( ν ) = π m ( ν ) = π ν and ν is divergen t for P j on lev el j . Consequently , the repair tick et issued in (I) on level j is not used in (I I) and, indeed, there are at least r j − 1 + 1 = r j a v ailable repair tic kets afterwards. – Assume that r j = r j − 1 > 0 and a repair tick et is used in (I I) on lev el j since π j ( ν ) 6 = π ν . Then ˜ q m j +1 ( ν ) = ˜ q m j ( ν ) and so | ˜ q i j +1 ( ν ) | = | ˜ q i j ( ν ) | . Consequently , j 6∈ ˜ q i j ( ν ) and hence ν is again divergen t for P j on level j . This means a repair tic ket for (I I) has just b een issued in (I) on level j . Claim (I II) is ﬁnished. Altogether, w e hav e issued at most b log 2 k c repair tick ets along P 0 j , and also at most b log 2 k c tic kets along P 00 j . F urthermore, t wo special repair tic kets are issued to µ j , summing up to at most 2 b log 2 k c + 2 = 2 b log 2 2 k c for iteration j . F or the whole problem, altogether at most 2 k b log 2 2 k c repair tic kets are issued. W e are hence nearly ﬁnished and it remains to prov e that the ab o ve distributed repair tic kets are suﬃcien t for all the “repair operations” carried out when drawing the edges of F into G 0 . Again for j = 1 , . . . , k (the order is now irrelev an t), we argue as follows. By Deﬁnition 3.18, the skeletons of α 0 j and µ j m ust share a vertex x j ∈ V ( G ) ( x j is a cut vertex of G if µ j 6 = α 0 j but this is not imp ortan t now). W e analogously set y j ∈ V ( G ) on the side of β 0 j . No w, P 0 j is the con-path of { u j , x j } and P 00 j is the con- path of { y j , v j } . By Claim 3.6(b), the preferences Π 0 j , which are the restriction of Π j to P 0 j , are optimal em b edding preferences of { u j , x j } . The em b edding G 0 hence honors Π 0 j with defect at most equal to the n umber of repair tick ets used along P 0 j —w e denote this num b er by t 0 j . W e deﬁne Π 00 j and t 00 j analogously . By Lemma 3.5, w e can draw the new edges u j x j , y j v j in to G 0 , constructing G 1 := G 0 + u j x j + y j v j with ins Σ ( G, { u j x j , y j v j } ) + ( t 0 j + t 00 j ) · b ∆ ( G ) / 2 c cross- ings. F urthermore, the new edge x j y j can b e drawn into G 1 with ins( G 1 , x j y j ) crossings—this ma y actually b e a nonzero num b er if µ j is an R-node, but there are no embedding preferences for such R-no des an ywa y , as their skele- ton em b eddings are not mutable. Altogether, G 2 := G 1 + x j y j is dra wn with ins Σ ( G, { u j x j , x j y j , y j v j } ) + ( t 0 j + t 00 j ) · b ∆ ( G ) / 2 c crossings. Using the same argument as in the pro of of Lemma 3.5, we no w p erturb the path formed by the edges u j x j , x j y j , y j v j in G 2 in to a drawing of the edge f j = u j v j in G 0 + f j . F or this, we require at most 2 · b ∆ ( G ) / 2 c additional crossings. The n umber of crossings in G 0 + f j is hence at most ins( G, f j ) + ( t 0 j + 2 + t 00 j ) · b ∆ ( G ) / 2 c using Claim 3.6(a). 32 While neglecting the at most  k 2  crossings b et w een the edges of F , we get that the total n umber of crossings b et ween G 0 and F is at most ins Σ ( G, F ) + k X j =1 ( t 0 j + 2 + t 00 j ) ·  ∆ ( G ) 2  ≤ ins Σ ( G, F ) +  ∆ ( G ) 2  · 2 k b log 2 2 k c . This concludes the pro of. u t 3.4 Tigh tness of the Analysis of Algorithm 3.7 Reading the ﬁne-grained analysis of Algorithm 3.7 in Theorem 3.19, it is natural to think whether p erhaps the ideas can b e improv ed further, giving an even tigh ter guaranteed relation b et ween the outcome of the algorithm and the sum of the individual insertion v alues ins Σ ( G, F ) than (7). This is, how ever, not p ossible as we no w show by exhibiting an asymptotically matching low er b ound in Prop osition 3.20. Prop osition 3.20. F or any inte gers r, k ≥ 0 and ∆ ≥ 4 , ther e exist instanc es of the MEI( G, F ) pr oblem such that k = | F | , ∆ ( G ) ≤ ∆ , ins Σ ( G, F ) ≥ r , and (8) ins( G, F ) ≥ ins Σ ( G, F ) + Ω  ∆ · k log 2 k +  k 2   wher e ins Σ ( G, F ) := P f ∈ F ins( G, f ) . Note, though, that this lo wer b ound only concerns the relation b etw een the optim um v alue ins( G, F ) and the simple low er b ound ins Σ ( G, F ) we use in our analysis; it do es not say anything ab out approximabilit y (or inapproximabilit y) of the MEI problem itself. The main message of this claim hence is that if one w ants to achiev e an algorithm with a tighter appro ximation guaran tee for the MEI problem, then one must consider something more than just the individual insertion solutions. Pr o of. W e are going to presen t three separate constructions. They can then b e easily com bined together by adjusting them appropriately to k and adding dumm y edges b et ween them (to satisfy connectivity of G ): I) There exists a planar graph G 1 and a vertex pair a, b ∈ V ( G 1 ) such that ins( G 1 , ab ) ≥ r and ∆ ( G 1 + ab ) = 3. I I) There exists a planar graph G 2 and a set of ` vertex pairs F 2 = { a i b i : a i , b i ∈ V ( G 2 ) , i = 1 , 2 , . . . , ` } , such that eac h G 2 + a i b i is planar, ∆ ( G 2 ) ≤ 4, and ins( G 2 , F 2 ) =  ` 2  . I II) There exists a planar graph G 3 and a set of m vertex pairs F 3 = { s i t i : s i , t i ∈ V ( G 3 ) , i = 1 , 2 , . . . , m } , such that each G 3 + s i t i is planar, ∆ ( G 3 ) ≤ ∆ , and ins( G 3 , F 3 ) ≥ 1 2 ∆ · m log 2 m . 33 s s 1 2 ... ... t 1 t 2 (a) In this picture  ` 2  crossings are required. 4 m m any (b) Graphs H 4 and H 6 ; there are 4 or 6 marked v ertices, 2 m disjoint (concentric) inner cycles, and eac h depicted line represents ∆/ 4 parallel edges. y x x ' y ' x " y " y x s " s ' (c) A detail of R-no de gadgets for (d), made of a cop y of H 6 and H 4 . x " y " x ' y ' x y (d) A scheme of the construction requiring 1 2 ∆ · m log 2 m crossings. Each shaded part in the picture is a copy of H 4 or H 6 with the depicted marked vertices. Fig. 3. Illustration of the pro of of Prop osition 3.20: constructions requiring man y cross- ings in an optimum MEI solution compared to individual edge insertions. In (I), we take as G 1 a suﬃciently large plane hexagonal grid, and select tw o edges in G 1 that are suﬃcien tly far from eac h other and from the grid boundary; the v ertices a, b then sub divide the selected edges. In (I I), we assume G 2 to b e any dense and large enough triconnected graph with a face of length 2 ` . Lab eling the vertices on this face s 1 , . . . s ` , t 1 , . . . t ` in clo c kwise order, gives an instance that clearly requires  ` 2  crossings. See Fig- ure 3(a) for an illustration. W e concen trate on the most interesting construction (I II). Since the b ound in (8) is of asymptotic nature, we may without loss of generality assume that ∆ is ev en and divisible by 4, and that m = 2 d for some in teger d ≥ 1. Let H q , q ∈ { 4 , 6 } , b e any dense enough planar graph with q sp ecial marke d v ertices v 1 , . . . , v q , eac h of degree exactly ∆/ 2, such that the follo wing conditions 34 hold. Up to mirroring and reordering of parallel edges, H q allo ws only a unique plane embedding of H q . Every such embedding H q 0 has a unique face, called active , inciden t with all of v 1 , . . . , v q in this order; no other face of H q 0 is inciden t with more than one of v 1 , . . . , v q . Moreo ver, assume we embed new b ar edges v 1 v 4 , v 2 v 3 , v 5 v 6 (only v 1 v 3 if q = 4) into the active face of H q 0 ; this divides the activ e face into 4 (or 2) se ctors . W e require that if we dra w any curve γ with the ends in distinct sectors and not crossing the bar edges, then γ mak es at least ∆/ 2 crossings with the edges of H q 0 . If the t wo sectors holding the ends of γ are not next to each other and γ av oids all other sectors, then γ makes at least m∆ crossings. The desired prop erties of H q are for instance ac hieved by the graphs depicted in Figure 3(b), where the marked v ertices are drawn white, the outer face is the activ e face, and each drawn line represents a bunch of ∆/ 4 parallel edges. Our recursive construction of G 3 for (I II) is sc hematically depicted in Fig- ure 3(d). Recalling m = 2 d , w e deﬁne a graph G s d with t wo special vertices called the p oles , b y induction on d . In the base case, d = 1, G s 1 is a copy of H 4 with the marked vertices x, s 00 , y , s 0 in this order around the active face, such that x, y are the p oles of G s 1 . Having constructed G s d and its disjoint copy G t d , we deﬁne G s d +1 as follows: take a copy B of H 6 (the b olt ) with the marked vertices x, x 00 , y 00 , y , y 0 , x 0 in this order (see Figure 3(c)), make x, y the new poles of G s d +1 , and iden tify x 0 , y 0 with the t wo p oles of G s d and x 00 , y 00 with the t wo p oles of G t d . The graph G s d has 2 d = m marked v ertices that are copies of s 0 , s 00 from G s 1 (the white terminals in Figure 3(c)), and we denote them by s 1 , . . . , s m (in any order). W e analogously denote by t 1 , . . . , t m the corresp onding vertices in G t d . Finally , we let G 3 := G s d +1 and F 3 = { s i t i : i = 1 , . . . , m } . It should be understo o d that the embedding p ossibilities for G 3 are essen tially deﬁned by the binary ﬂipping decisions of each R-no de skeleton corresp onding to a copy of H 4 or H 6 in the construction. F urthermore, for each 1 ≤ i ≤ m , the edge s i t i can b e inserted into (some embedding of ) G 3 without crossings and into any ﬁxed embedding of G 3 with at most d∆ crossings. See the red dotted edges depicted in Figure 3(d). T o settle (I II) it remains to argue that ins( G 3 , F 3 ) ≥ 1 2 ∆ · md , whic h we achiev e in tw o steps. Let ins 0 ( G 3 , F 3 ) denote the solution v alue of the MEI( G 3 , F 3 ) problem without counting the crossings b et w een edges of F 3 . Claim 3.21. L et G 3+ b e the gr aph obtaine d fr om G 3 by adding a bunch F + of ∆/ 2 p ar al lel e dges b etwe en the two p oles of G 3 (cf. the blue dashe d curve in Figur e 3(d)). Then ins 0 ( G 3+ , F 3 ) = ins 0 ( G 3 , F 3 ) + m∆/ 2 . W e ha ve ins( G 3+ , F 3 ) ≤ ins( G 3 , F 3 ) + m∆/ 2 since b oth p oles of G 3 share a com- mon face ϕ in the corresp onding optimal solution and each edge of F 3 tra verses an y face of embedded G 3 at most once. Routing the edges of F + via ϕ hence giv es at most ∆/ 2 additional crossings p er edge of F 3 . Conv ersely , consider an optimal solution of v alue ins 0 ( G 3+ , F 3 ) and an edge s i t i ∈ F 3 . W e claim that after remo ving F + from this solution, w e “sav e” at least ∆/ 2 crossings on s i t i . If s i t i crossed all of F + , then w e are done. Otherwise, by the property of H 6 , the drawing of s i t i has to enter b oth the sectors of the b olt of G 3 inciden t to 35 the p oles (and so has additional ∆/ 2 crossings in b et w een), or there are at least m∆ > d∆ + ∆/ 2 crossings on s i t i . In b oth cases one requires by at least ∆/ 2 less crossings on s i t i in the sub em b edding of G 3 . Claim 3.22. ins 0 ( G 3 , F 3 ) ≥ 1 2 ∆ · md and ins 0 ( G 3+ , F 3 ) ≥ 1 2 ∆ · m ( d + 1) . By Claim 3.21, it is enough to prov e one of the inequalities but it is conv enien t to consider b oth of them by indunction on d ≥ 0. The base case of G 3 = G s 1 ( d = 0) is a degenerate extension of the previous, deﬁning s 1 , t 1 to b e the tw o opp osite marked vertices s 0 , s 00 of H 4 , and it is trivial. Stepping from d − 1 to d ≥ 1; we consider G 3 = G s d +1 and an optimal solution of v alue ins 0 ( G 3 , F 3 ) inserting the m edges of F 3 in to a plane embedding G 1 of G 3 , without counting crossings inside F 3 . Let β b e a curv e drawn b et ween the p oles of G 1 without crossing its edges (e.g., the blue dashed curv e in Figure 3(d)). Similarly as in the pro of of Claim 3.21, we can show that there is an optimal solution in whic h every edge of F 3 has to cross β . Let G 0 0 and G 00 0 b e the sub em b eddings of the t wo recursive copies of G s d in G 1 . W e cut every curve of the edges of F 3 at a p oin t in the intersection with β , to obtain 2 m curv e parts, m of which are incident to G 0 0 . These latter parts can b e paired such that the pairs (after reconnection along β ) form a set of m/ 2 edges F 0 3 with ends in V ( G 0 0 ) and the instance ( G 0 0 , F 0 3 ) is as in the inductive assumption for d − 1. By the prop ert y of H 6 , the b olt of G 1 can play the role of F + in Claim 3.21; let G 0 0 + denote G 0 0 augmen ted with the b olt. W e hence hav e that the curve parts of F 3 inciden t to G 0 0 ha ve at least ins 0 ( G 0 0 + , F 0 3 ) crossings with edges of G 1 b y our (second claim of the) inductive assumption. Since the same symmetrically holds for the curve parts incident to G 00 0 , summing together w e get that ins 0 ( G 3 , F 3 ) ≥ 2 · ins 0 ( G 0 0 + , F 0 3 ) ≥ 2 · 1 2 ∆ · m/ 2 · ( d − 1 + 1) = 1 2 ∆ · md , as desired. u t R emark 3.23. Notice that the actual MEI instances constructed in the pro of of Prop osition 3.20 are not at all “bad” for us—they will alwa ys b e solv ed to optimalit y in Algorithm 3.7. The example is thus if purely theoretical nature. 3.5 Run time of Algorithm 3.7 Thanks to using well-kno wn building blo c ks, the ov erall runtime b ound of our algorithm is actually rather simple to see. Let V = V ( G ). As mentioned in Section 2, we can build the con-tree (step (1)) in linear time O ( | V | ), based on the linear-time decomp osition algorithm [22]. Recall that | V ( C ) | = O ( | V | ). In step (2), w e call the (deterministic) O ( | V | ) insertion algorithm k times. In step (3), computing p ( ν ) for all ν ∈ V ( C ) is trivial in O ( k | V | ), as is ac hieving the weak estimate b y setting, e.g., p 0 ( ν ) = p ( ν ). T o achiev e the strong estimate, we can compute a simplicial sequence (Claim 3.15) via k BFS-tra versals in O ( k | V | ) time. Thereby , we also identify no de substitutes (Lemma 3.17) and decide whic h preferences to ignore (Deﬁnition 3.18). 36 In step (4), we can compute π ν for each of the con-tree no des ν via semi- ma jority vote in O ( k ) time. Computing the embedding G 0 based on these pref- erences takes only linear time O ( | V | ). Hence, also this step requires O ( k | V | ) time. In step (5), we then run k (deterministic) BFS algorithms, requiring O ( | V | ) time each. Since each edge has at most O ( | V | + k ) crossings in the end, the realization may require up to O ( k | V ( G ) | + k 2 ) time, which hence constitutes the o verall runtime b ound of the algorithm, as given in Theorem 1.2. 4 Crossing Num b er Approximations Our main concept of in terest is the crossing num b er of the graph G + F . W e can com bine our ab ov e result with a result of [9], connecting the optimal crossing n umber with the problem of multiple edge insertion. Theorem 4.1 (Chimani et al. [9]). Consider a planar gr aph G and an e dge set F , F ∩ E ( G ) = ∅ . The value ins( G, F ) of an optimal solution to MEI( G, F ) satisﬁes ins( G, F ) ≤ 2 | F | ·  ∆ ( G ) 2  · cr( G + F ) +  | F | 2  wher e cr( G + F ) denotes the (optimal) cr ossing numb er of the gr aph G including the e dges F , and  | F | 2  ther eby ac c ounts for cr ossings b etwe en the e dges of F . Notice that, when considering the crossing n umber problem of G + F , w e ma y assume G to b e connected—otherwise w e could “shift” some edges of F to G . Let k = | F | , ∆ = ∆ ( G ). Plugging the estimate of Theorem 4.1 into the place of ins Σ ( G, F ) ≤ ins( G, F ) in Theorem 3.19, and realizing that the  k 2  term in b oth estimates stands for the same set of crossings, we immediately obtain ins aprx ( G, F ) ≤ 2 k · b ∆/ 2 c · cr( G + F ) + 2 k b log 2 k cb ∆/ 2 c +  k 2  Hence we can give the outcome of Algorithm 3.7 as an approximate solution to the crossing n umber problem on G + F , proving: Theorem 4.2 (Theorem 1.2(5)). Given a planar gr aph G with maximum de gr e e ∆ and an e dge set F , | F | = k , F ∩ E ( G ) = ∅ , Algorithm 3.7 c omputes, in O ( k · | V ( G ) | + k 2 ) time, a solution to the cr( G + F ) pr oblem with the fol lowing numb er of cr ossings cr aprx ( G + F ) ≤ b ∆/ 2 c · 2 k · cr( G + F ) + 2 k b log 2 k c · b ∆/ 2 c + 1 2  k 2 − k  . u t 37 A note on appr oximating the cr ossing numb er of surfac e-emb e dde d gr aphs. In [19], an algorithm is presented to appro ximate the crossing num b er of graphs embed- dable in an y ﬁxed higher orien table surface. This algorithm lists the technical requiremen t that G has a “suﬃciently dense” embedding on the surface. Y et, as noted in [19], a result like Theorem 4.2 allows to drop this requirement: If the embedding density is small, then the remov al of the oﬀending small set(s) of edges is suﬃcient to reduce the graph gen us, while the remov ed edges can b e later inserted in to an intermediate planar subgraph of the algorithm. 5 A Note on the Planarization Heuristic and the Practicalit y of our Algorithm The curren tly practically strongest heuristic [17] for the crossing num b er problem is the planarization heuristic whic h starts with a maximal planar subgraph of the giv en non-planar graph, and then iterativ ely performs single edge insertions. The crossings of such an insertion are then replaced by dumm y no des such that each edge is inserted into a planar graph. Due to its practical sup erior p erformance, often giving the optimal solution [5, 15], it was an op en question if this approach unkno wingly guarantees some approximation ratio. By inv estigating our strategy and pro ofs, it b ecomes clear that this approac h as such cannot directly giv e an approximation guarantee: by routing an edge (in an R-no de) through another virtual edge (representing a subgraph S ) and replacing the crossings with dummy no des, you essentially ﬁx (most of ) the em b edding of S . This ﬁx might result in O ( n ) em b edding restrictions for fur- ther edge insertions, without having an edge in F that requires this embedding. Therefore the num ber of dirt y passes can no longer b e bounded b y a function in k . Y et, an implemen tation realizing the planarization heuristic already con tains all the ingredien ts to obtain our approximation; one “only” has to compute all the embedding preferences and merge them according to Algorithm 3.7, steps (3) and (4), b efore running the ﬁxed-embedding edge insertion subalgorithm for all inserted edges. In fact, such an implementation is describ ed in [6], where, in a nutshell, it is sho wn that this algorithm is m uch faster then the b est p ostprocessing-heavy planarization-based heuristics, while pro ducing roughly equally go od solutions. 6 Conclusions W e hav e presented a new approximation algorithm for the m ultiple edge insertion problem which is faster and simpler that the only formerly known one [11], while at the same time giving better b ounds; in fact, in con trast to the former m ultiplicative approximation, it is the ﬁrst one with an additive b ound. Our algorithm directly leads also to improv ed approximations (ev en with constant ratio for a large class of inputs) for the crossing num b er problem of graphs in 38 whic h a given set of edges can b e remov ed in order to obtain a planar subgraph, and for graphs that can b e embedded on a surface of some ﬁxed genus. W e conclude with an interesting open problem. 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Crossing num b ers of graphs: A bibliography . ftp://ftp.ifi.savba.sk/ pub/imrich/crobib.pdf , 2011. 26. T. Ziegler. Cr ossing Minimization in Automatic Gr aph Dr awing . PhD thesis, Saarland Univ ersity , Germany , 2001. 40 A Ziegler’s pro of of NP-hardness of MEI In his PhD-thesis [26], Ziegler sho wed that MEI is NP-hard (the corresp onding decision problem is NP-complete) for general k . Since this thesis is somewhat hard to obtain, w e repro duce a sligh tly simpliﬁed version of his pro of. Theorem A.1 (Ziegler [26]). Given a gr aph G , a set F of unor der e d vertex- p airs, and an inte ger b , it is NP-c omplete to de cide whether ther e is a planar dr awing D of G such that we c an insert an e dge v w for e ach vertex p air { v , w } ∈ F into D with over al l at most b cr ossings. Pr o of. NP-membership is trivial and it hence remains to show NP-hardness. W e use a reduction from Fixed Linear Crossing Number (FLCN): Giv en a graph H = ( V , E ), a 1-to-1 function f : V → { 1 , 2 , . . . , | V |} , and an integer ` . Do es there exist an f -linear drawing of H with at most ` crossings? Thereb y , an f -line ar dr awing is one where all v ertices are placed on a horizontal line, each vertex v at co ordinate f ( v ), and each edge is either drawn completely ab o v e or completely b elow that line. It w as sho wn in [23] that Flcn is NP- complete. Let ( H = ( V , E ) , f , ` ) b e an instance to FLCN. W e will construct a corre- sp onding MEI instance ( G = ( W , E ∗ ) , F , b ) of size p olynomial in | V ( H ) + E ( H ) | whic h is a yes-instance for MEI if and only if ( H, f , ` ) is a yes-instance for FLCN. The key idea is to build a rigid graph G that mo dels the restrictions of f -linear dra wings, into which we then hav e to insert the original edges E . Observe that FLCN can only b e hard if ` < | E | 2 . Lab el the v ertices V = { v 1 , . . . , v n } suc h that f ( v i ) = i and n = | V | . W e may assume w.l.o.g. that n ≥ 3. W e start with constructing a graph G 0 = ( W, E 0 ) on n + 2 vertices where W := V ∪ { w a , w b } and E 0 := { v i v i +1 : 1 ≤ i < n } ∪ { v 1 w a , v n w a , v 1 w b , v n w b , w a w b } . Observe that G 0 is planar and (since its SPR-tree consists of one R- and one S-no de) allows only a unique embedding (and its mirror). W e obtain G from G 0 b y replacing each edge in E 0 b y | E | 2 parallel edges. Up to ordering the multiple edges amongst its peers, G still allows only a unique em b edding. No w, set b := ` and F := E , i.e., w e wan t to insert the edges of H into planar G . W e can assume w.l.o.g. that E contains no edges v i v i or v i v i +1 for an y i . G has exactly four faces with more than t wo incident vertices: let ϕ 1 ( ϕ 2 ) b e the face incident to exactly { w a , w b , v 1 } ( { w a , w b , v n } , resp ectiv ely). Let ϕ a ( ϕ b ) b e the face incident to all of V and w a ( w b , resp ectively). T o go from one of these four faces to another, we w ould alwa ys hav e to cross at least | E | 2 edges (a parallel bunch), which is infeasible when asking for a solution with at most b = ` < | E | 2 crossings. No edge of F = E will b e placed in ϕ 1 or ϕ 2 as they are only incident to one vertex of V . Hence each edge is either completely within ϕ a or ϕ b , and the equiv alence with b eing ab o ve or b elo w the horizon tal line of an f -linear drawing follows. u t 41

A Tighter Insertion-based Approximation of the Crossing Number

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