Cubicity, Degeneracy, and Crossing Number

Cubicit y , Degeneracy , and Crossing Num b er Abhijin Adiga, L. Sunil Cha ndran, and Roger s Mathew Department of Computer Scie n ce and Automation, Indian Institute of Science, Bangalore - 560012, India. { abhijin, sunil,rogers } @csa.iisc.ernet.in Abstract. A k - box B = ( R 1 , . . . , R k ), where each R i is a clos ed interv al on the real line, is defined to b e th e Cartesi an pro duct R 1 × R 2 × · · · × R k . If eac h R i is a unit length in terva l, we call B a k - cube. Boxicity of a graph G , denoted as box ( G ), is the min imum integer k such that G is an interse ct ion graph of k -b o x es. Similarly , the cubicity of G , den oted as cub ( G ), is the minimum integer k such that G is an in tersection graph of k -cub es. It wa s shown in [L. Sunil Chandran, Mathew C. F rancis, and Nav een Siv adasan: Rep resen ting graphs as the in tersection of axis-parallel cub es. MCDES-2008, IISc Centenary Confer enc e , a v ailable at CoRR , abs/cs/ 0607092 , 2006.] that, for a graph G with max imum degree ∆ , cub ( G ) ≤ ⌈ 4( ∆ + 1) log n ⌉ . I n this pap er, we sho w that, for a k -degenerate graph G , cub ( G ) ≤ ( k + 2) ⌈ 2 e log n ⌉ . Since k is at most ∆ and can be m u c h lo wer, this clearly is a stronger result. This bound is ti ght. W e also give an efficient deterministic algo rith m that runs in O ( n 2 k ) t ime to output a 8 k ( ⌈ 2 . 42 log n ⌉ + 1) dimensional cub e rep resentation for G . The cr ossing numb er of a graph G , denoted a s C R ( G ), is the minimum num b er of cross ing pairs of edges, ove r all drawings of G in the plane. An imp ortant consequence of the ab ov e result is that if the crossing number of a graph G is t , then box ( G ) is O ( t 1 4 ⌈ log t ⌉ 3 4 ) . This bound is tigh t up to a factor of O ((log t ) 1 4 ). W e also show that, if G has n vertices , then cub ( G ) is O (log n + t 1 / 4 log t ). Let ( P , ≤ ) b e a partially ordered set and let G P denote its u nderlying comparabilit y graph. Let dim ( P ) denote the p oset dimension of P . An- other interesting consequence of our result is to show that dim ( P ) ≤ 2( k + 2) ⌈ 2 e log n ⌉ , where k denotes the degeneracy of G P . Also, w e get a deterministic algori t h m that ru ns in O ( n 2 k ) time to construct a 16 k ( ⌈ 2 . 42 log n ⌉ + 1) sized realizer for P . A s far as we k no w, though very goo d u pper b ounds exist for p oset dimension in t erms of maximum de- gree of its underlying comparability graph, no upp er boun d s in terms of the degeneracy of th e u nderlying comparability graph is seen in the literature. It wa s shown in [L. Sunil Chandran, Mathew C. F rancis, and Nav een Siv adasan: Geometric R epresen tation of Graphs in Lo w Dimension Us- ing Axis P arallel Boxes. Algori t h mica 56(2): 129-140, 2010. ] that boxicit y of almost all graphs in G ( n, m ) mo d el is O ( d av log n ), where d av = 2 m n denotes the av erage degree of the graph un d er consideration. In this pap er, w e prov e a stronger result. U sing ou r b ound for the cubicity of k - degenerate graphs, we sho w that cubicit y of almost all graphs in G ( n, m ) mod el is O ( d av log n ). Keywords: Degeneracy , Cubicity , Bo x icit y , Crossing Nu m b er, Inter- v al Graph, I n tersection Graph, Poset Dimension, Comparability Graph , random graph, avera ge degree 1 In tro duction A g raph G is an interse ction gr aph of sets fr om a family of sets F , if ther e exists f : V ( G ) → F such that ( u , v ) ∈ E ( G ) ⇔ f ( u ) ∩ f ( v ) 6 = ∅ . Representations of graphs a s the int er section gr a phs of v ar io us geometrical o b jects is a well s tudied topic in gra ph theory . Probably the most well studied class of intersection gra phs are the interval gr aphs . Interv a l graphs are the intersection graphs of closed int er v als on the real line. A res tricted form of in terv al graphs, that a llo w only int er v als of unit length, are indiffer enc e gr aphs or un it interval gr aphs . An interv al on the r eal line can b e gener a lized to a “ k -b o x ” in R k . A k -b o x B = ( R 1 , . . . , R k ), wher e e ac h R i is a closed in ter v al on the real line, is defined to b e the Cartesian pro duct R 1 × R 2 × · · · × R k . If eac h R i is a unit length int er v al, we call B a k -cub e. Thus, 1- b oxes are just closed in ter v als on the real line whereas 2- boxes are axis-parallel r ectangles in the pla ne. The parameter boxicity o f a gr aph G , denoted as box ( G ), is the minim um in teg er k such that G is an in ter section gra ph of k -b oxes. Similarly , the cubicity o f G , denoted as cub ( G ), is the minim um in teger k such that G is an intersection graph of k - cube s . Thus, interv al graphs are the g raphs with boxicit y equal to 1 a nd unit int er v al gr aphs ar e the gra phs with cubicity e qual to 1 . A k -b ox r epr esentation or a k dimensional b ox r epr esentation o f a g raph G is a mapping of the vertices of G to k -b oxes such that t wo vertices in G ar e adjacent if and o nly if their corres p onding k -b oxes hav e a non-empt y in ters ection. In a s imilar w ay , we define k -cub e r epr esent ation (or k dimensional cub e r epr esentation ) of a g raph G . Since k -cubes by definition are also k -boxes, b oxicity of a graph is a t most its cubicity . The concepts of b oxicit y and cubicity were introduced by F.S. Rob erts in 1969 [19]. Ro berts showed that for any gra ph G o n n vertices box ( G ) ≤ ⌊ n 2 ⌋ and cub ( G ) ≤ ⌊ 2 n 3 ⌋ . Both these b ounds are tight since box ( K 2 , 2 ,..., 2 ) = ⌊ n 2 ⌋ and cub ( K 3 , 3 ,..., 3 ) = ⌊ 2 n 3 ⌋ where K 2 , 2 ,..., 2 denotes the complete n/ 2-pa rtite graph with 2 vertices in eac h part and K 3 , 3 ,..., 3 denotes the complete n/ 3 -partite g r aph with 3 vertices in each part. It is easy to see that the boxicity of any gra ph is at least the b oxicit y of a ny induced subgra ph o f it. Box represe n tation of gra phs finds a pplication in nic he ov er lap (comp etition) in ecology and to problems of fleet maintenance in oper ations research (see [10]). Given a low dimensional box representation, some well known NP-hard pr oblems bec ome p olynomial time solv able. F o r instance, the ma x-clique problem is po ly- nomial time so lv able for graphs with b o xic it y k b ecause the num b er of maxima l cliques in such gr aphs is o nly O ((2 n ) k ). 2 1.1 Previous R esults on Bo xicity and Cubicit y It was sho wn by Cozzens [9] that computing the b oxicity of a graph is NP -ha rd. Krato ch v ´ ıl [14] s ho wed that deciding whether the b oxicit y o f a gra ph is at most 2 itself is NP -co mplete. It has b een s ho wn by Y a nna k ak is [23 ] tha t deciding whether the c ubicity o f a given graph is at lea st 3 is NP -hard. Researchers hav e tried to b ound the boxicity and cubicity of graph c la sses with sp ecial structure. Scheinerman [20] s howed that the b oxicit y of o uterplanar graphs is at most 2. Thomassen [2 1 ] prov ed that the b oxicit y of plana r graphs is bo unded fro m a bov e by 3. Upp er b ounds for the boxicity of ma ny o ther gr aph classes such as chordal gr aphs, A T-fr e e graphs, p ermutation graphs etc. were shown in [8 ] by relating the b oxicit y of a graph with its treewidth. The cub e representation o f sp ecial classes of graphs lik e hyper cubes a nd co mplete m ulti- partite graphs were investigated in [19, 1 5, 16]. V arious other upp er b ounds on b oxicit y a nd c ubicit y in terms of graph pa - rameters s uc h as maximum degree, treewidth etc. can be seen in [5, 3, 4, 12 , 8 ]. The ratio o f cubicity to b oxicit y of any gr aph on n vertices was s ho wn to b e at most ⌈ log 2 n ⌉ in [6]. 1.2 Equiv alen t Defin i tions for Bo xi ci ty and Cubicity Let G a nd G 1 , ..., G b be graphs such that V ( G i ) = V ( G ) for 1 ≤ i ≤ b . W e say G = T b i =1 G i when E ( G ) = T b i =1 E ( G i ). Belo w, we sta te tw o very useful lemmas due to Rob erts [19 ]. Lemma 1. F or any gr aph G , box ( G ) ≤ k if and only if ther e exist k interval gr aphs I 1 , . . . , I k such that G = I 1 ∩ · · · ∩ I k . Lemma 2. F or any gr aph G , cub ( G ) ≤ k if and only if ther e exist k indiffer enc e gr aphs (unit interval gr aphs) I 1 , . . . , I k such that G = I 1 ∩ · · · ∩ I k . 1.3 Our Res ults A gra ph G is k -de gener ate if the vertices o f G can b e enumerated in s uc h a wa y that every vertex is succeeded by at most k of its neighbor s. The least nu mber k such that G is k -degenerate is called the deg eneracy o f G and a n y such enumeration is referr ed to as a de gener acy or der of V ( G ). F or example, trees a nd forests ar e 1-degenerate a nd planar g r aphs are 5- deg enerate. Series- parallel gra phs, outerpla nar graphs, non-regula r cubic g raphs, circle graphs of girth at lea st 5 etc. are sub classes o f 2 -degenerate gra phs. Main Re sult: It was sho wn in [3] that, for a graph G with maximum deg ree ∆ , c u b ( G ) ≤ ⌈ 4( ∆ + 1) log n ⌉ . In this paper , we show that, for a k -deg enerate graph G , cub ( G ) ≤ ( k + 2) ⌈ 2 e log n ⌉ . Since k is at most ∆ and can be m uch lower, this clearly is a stro nger result. W e prove that this b ound is tig ht. Mo reov er, we give an efficient deterministic algorithm that outputs a 8 k ( ⌈ 2 . 42 lo g n ⌉ + 1) dimensional cub e represe ntation for G in O ( n 2 k ) time. 3 Consequence 1: The cr ossing nu mb er of a gra ph G , denoted as C R ( G ), is the minim um num b er of cro ssing pairs of edges, ov er all drawings of G in the plane. W e prov e that, if C R ( G ) = t , then box ( G ) ≤ 66 t 1 4 ⌈ log 4 t ⌉ 3 4 + 6. This bo und is tight up to a facto r of O ((log t ) 1 4 ). W e also show that, if G has n vertices, then cu b ( G ) is O (log n + t 1 / 4 log t ). See Section 5 fo r details. Consequence 2 : It was shown in [5] that b oxicit y o f almost all graphs in G ( n, m ) mo del is O ( d av log n ), where d av = 2 m n denotes the av er age degr ee of the g r aph under considera tio n. What can we infer ab out the cubicity o f almost all graphs from the result of [5]? It was shown in [6] that for every g raph G , cub ( G ) ≤ lo g 2 n × box ( G ). Co mbining this result with that of [5], we can infer that cubicity of almost a ll g raphs is O ( d av log 2 n ). In this pap er, we prove a stronger result. Using o ur b ound for the cubicity of k -degene r ate gra phs, we show that cubicity of almost all graphs in G ( n, m ) mo del is O ( d av log n ). See Section 6 for de ta ils. Consequence 3: Let ( P , ≤ ) be a p oset (partia lly ordered set) and let G P be the underlying compar abilit y gr aph of P . A linear extens io n L of P is a total order which satisfies ( x ≤ y ∈ P ) = ⇒ ( x ≤ y ∈ L ). A rea lizer of P is a set of linear e xtensions of P , s a y R , which satisfy the following condition: for a n y t wo distinct e le men ts x and y , x ≤ y in P if and only if x ≤ y in L , ∀ L ∈ R . The p oset dimension o f P , denoted by dim ( P ), is the minim um integer k such that there exis ts a realizer o f P of cardina lit y k . Y a nnak a kis [23 ] show ed that it is NP-complete to decide whether the dimension o f a p oset is at most 3. The p oset dimension is an extensively studied parameter in the theor y o f partial order (See [22] for a co mprehensiv e treatment). There a re se veral re s earch pap ers in the partial order literature which study the dimension of p osets whose under lying co mparabilit y gr aph has s ome sp ecial structure – in terv al or de r , semi order and cr o wn p osets ar e some examples. While very go o d upp er b ounds (for example c∆ (log ∆ ) 2 , whe r e c is a consta n t) a r e known for p oset dimension in ter ms of maximum degr ee ∆ of its underlying comparability gra ph, as far as we know there a re no upp er b ounds in ter ms of the deg eneracy of the underlying comparability gra ph. Co nnecting our main result with a result in [1], we can get an upper b ound for p oset dimensio n in terms of the degener acy of the under ly ing compa rabilit y gra ph as follows. It was shown in [1] that dim ( P ) < 2 box ( G P ). Therefor e, if the deg eneracy of the underlying comparability gr a ph G P is k , then o ur result says that dim ( P ) ≤ 2( k + 2) ⌈ 2 e log n ⌉ . Also, we get a deterministic algo rithm that runs in O ( n 2 k ) time to co nstruct a 1 6 k ( ⌈ 2 . 42 log n ⌉ + 1) s iz ed realiz er for P . 2 Preliminaries F or any finite p ositive integer n , let [ n ] deno te the se t { 1 , . . . n } . Unless men tio ned explicitly , all logarithms a re to the bas e e in this paper . All the graphs that we consider ar e simple, finite and undirected. F or a g raph G , w e denote the v er tex set of G by V ( G ) a nd the edg e set of G by E ( G ). F or any v er tex u ∈ V ( G ), 4 N G ( u ) = { v ∈ V ( G ) | ( u , v ) ∈ E ( G ) } . W e define d G ( u ) := | N G ( u ) | . T he average degree of G is denoted by d av ( G ). Consider a g raph G who se vertices are par titio ne d into tw o pa rts, namely V A and V B . That is, V ( G ) = V A ⊎ V B . W e shall use S B ( G ) to denote the graph with V ( S B ( G )) = V ( G ) and E ( S B ( G )) = E ( G ) \ { ( u, v ) | u, v ∈ V B } . In other words, S B ( G ) is o bta ined from G by making V B a stable set (or an independent set). Let C B ( G ) denote the graph with V ( C B ( G )) = V ( G ) and E ( C B ( G )) = E ( G ) ∪ { ( u, v ) | u, v ∈ V B } . That is, C B ( G ) is obtained from G by making V B a clique. L e t G B denote the subgraph of G induced on V B . Analogously , we define S A ( G ) , C A ( G ), and G A . Since an interv al g raph is the intersection g r aph of closed in ter v als on the real line, for ev er y interv a l gra ph I a , there exists a function f a : V ( I a ) → { X ⊆ R | X is a closed interv al } , such that ∀ u , v ∈ V ( I a ), ( u , v ) ∈ E ( I a ) ⇔ f a ( u ) ∩ f a ( v ) 6 = ∅ . The function f a is called a n interval r epr esentation of the in ter- v al g raph I a . No te that the interv al representation of an interv al gr aph need not be unique. In a similar way , we call a function f b a u nit interval r epr esentation o f unit in terv al graph I b if f b : V ( I b ) → { X ′ ⊆ R | X ′ is a unit leng th clo sed interv al } , such that ∀ u, v ∈ V ( I b ), ( u, v ) ∈ E ( I b ) ⇔ f b ( u ) ∩ f b ( v ) 6 = ∅ . Given a closed in- terv al X = [ y , z ], we define l ( X ) := y and r ( X ) := z . W e say that the interv a l X has left end-p oint l ( X ) a nd right end-p oint r ( X ). Given a graph G , a c oloring C of V ( G ) using colo rs χ 1 , . . . , χ a is a map C : V ( G ) → { χ 1 , . . . , χ a } . F o r each u ∈ V ( G ), we shall use C ( u ) to denote the color of u in C . Definitions, Notations and Assum ptions u s ed in Sections 3 and 4: Re- call that the degener acy of a g r aph is the lea st nu mber k s uc h tha t it has a vertex enum e ration in which e a c h vertex is succe e ded by at most k of its neigh- bo rs. Such a n enumeration is ca lle d the deg eneracy order . The gra ph G that we consider in these s ections is a k -degener a te gra ph having V ( G ) = { v 1 , . . . , v n } , | E ( G ) | = m and m (=  n 2  − m ) denotes the num b er of non-edges in G . The enu- meration v 1 , . . . , v n is a degeneracy order of V ( G ) and is denoted by D . F or every v i , v j ∈ V ( G ), we say v i < D v j if v i comes b efore v j in D i.e., v i < D v j if and only if i < j . Supp ose v i < D v j . I f ( v i , v j ) ∈ E ( G ), then we call v j a forwar d neighb or of v i and v i is refer r ed to as a b ackwar d neighb or o f v j . Obser v e that s ince G is k -degenerate, a vertex can hav e at most k forward neighbo rs. If ( v i , v j ) / ∈ E ( G ), then v j a forwar d non-neighb or of v i and v i is a b ackwar d non-n eigh b or of v j . F or any u ∈ V ( G ), N f G ( u ) = { w ∈ V ( G ) | w is a for w ar d neighbor of u } and N b G ( u ) = { w ∈ V ( G ) | w is a backward neighbor of u } . Supp ort s ets of a non-e dg e: F or each ( v x , v y ) / ∈ E ( G ), where v x < D v y , let S xy = { v z ∈ N f G ( v x ) | v y < D v z } ∪ { v y } . W e ca ll S xy the we ak supp ort set of the non-edge ( v x , v y ). Define T xy = S xy ∪ { v x } . W e call T xy the s t r ong supp ort set of the non-edg e ( v x , v y ). Let C b e a coloring (need not b e pro per) o f V ( G ). W e say S xy is favor ably c olor e d in C , if C ( v y ) 6 = C ( v w ), ∀ v w ∈ S xy \ { v y } . W e say T xy is favor ably c olor e d in C , if C ( v y ) 6 = C ( v w ), ∀ v w ∈ T xy \ { v y } 5 3 Cub e Represen tation and Coloring Lemma 3. L et G b e a k -de gener ate gr aph. L et χ = { χ 1 , . . . χ a } b e a set of c olors and let C = {C 1 , . . . , C b } b e a family of c olorings (ne e d not b e pr op er) of V ( G ) , wher e e ach C i uses c olors fr om the set χ . If the str ong su pp ort set T xy of every non- e dge ( v x , v y ) / ∈ E ( G ) , v x < D v y , is favor ably c olor e d in some C i , wher e i ∈ [ b ] , then cub ( G ) ≤ ab . Pr o of. W e pr ove this by construc ting ab unit interv al g raphs I i,j on the vertex set V ( G ), wher e i ∈ [ a ] and j ∈ [ b ], such that G = T a i =1 T b j =1 I i,j . Then the statement will follow from Lemma 2. Let f i,j denote a unit interv al representa- tion of I i,j . Let us par tition the v er tice s of I i,j int o t wo pa r ts, namely A ij and B ij , where A ij = { v ∈ V ( G ) | C i ( v ) = χ j } and B ij = V ( G ) \ A ij . F or every i ∈ [ a ] and j ∈ [ b ], a unit interv al representation f i,j of I i,j is constructed from the color ing C i in the following way . F or every v y ∈ V ( G ), If v y ∈ A ij , then f i,j ( v y ) = [ y + n, y + 2 n ] else f i,j ( v y ) = [ g ij max ( v y ) , g ij max ( v y ) + n ], wher e g ij max ( v y ) = max( { g | ( v y , v g ) ∈ E ( G ) , v g ∈ A ij } ∪ { 0 } ) . Since the length of f i,j ( v y ) is n , for every v y ∈ V ( G ), I i,j is a unit interv al graph. It is easy to see that, ∀ v x , v y ∈ A ij , 2 n ∈ f i,j ( v x ) ∩ f i,j ( v y ) a nd therefore A ij forms a clique in I i,j . Since n ∈ f i,j ( v x ) ∩ f i,j ( v y ), ∀ v x , v y ∈ B i,j , B i,j to o forms a clique in I i,j . F or every ( v x , v y ) ∈ E ( G ), with v x ∈ A ij and v y ∈ B ij , we hav e l ( f i,j ( v y )) = g ij max ( v y ) ≤ n ≤ l ( f i,j ( v x )) = n + x ≤ n + g ij max ( v y ), where the last inequality is inferred from the fact that ( v x , v y ) ∈ E ( G ) and v x ∈ A ij . But n + g ij max ( v y ) = r ( f i,j ( v y )). Therefore , we get l ( f i,j ( v y )) ≤ l ( f i,j ( v x )) ≤ r ( f i,j ( v y )) and hence ( v x , v y ) ∈ E ( I i,j ). Hence I i,j is a sup ergra ph o f G . Let v x < D v y and ( v x , v y ) / ∈ E ( G ). W e now hav e to show that ther e exists some unit interv a l gr a ph I i,j such that ( v x , v y ) / ∈ E ( I i,j ). W e know that, by assumption, ther e exists a coloring, say C i (where i ∈ [ a ]), such that the strong suppo rt se t T xy is fa vorably co lored in C i . Let χ j = C i ( v y ). Let g = g ij max ( v x ). W e claim that g < y . Assume, for con tr adiction, that g > y . Then g 6 = 0 and v g ∈ A ij . Since y > x , we g et g > x . Therefor e, v g ∈ N f G ( v x ) a nd g > y . This implies that v g ∈ T xy . Since T xy is fav o rably colored in C i , C i ( v g ) 6 = χ j . This contradicts the fact that v g ∈ A ij . Thus we prove the c la im. Therefo r e, r ( f i,j ( v x )) = n + g < n + y = l ( f i,j ( v y )) and hence ( v x , v y ) / ∈ E ( I i,j ). W e infer that G = T a i =1 T b j =1 I i,j . R emark 1. Note that ∀ v ∈ V ( G ) , i ∈ [ a ] , j ∈ [ b ], either f i,j ( v ) ∩ [ n, n ] 6 = ∅ or f i,j ( v ) ∩ [2 n , 2 n ] 6 = ∅ or b oth. ⊓ ⊔ 6 4 Cubicit y and D egeneracy 4.1 An Upp er Bound - Probabili stic Approac h Theorem 1. F or every k -de gener ate gr aph G , cub ( G ) ≤ ( k + 2) · ⌈ 2 e log n ⌉ Pr o of. Let χ = { χ 1 , . . . , χ k +2 } b e a set of k + 2 colors . Generate a rando m coloring C 1 (need no t be a pro per color ing) of vertices of G in the following w ay: F or each vertex v x ∈ V ( G ), pick a c o lor χ j , where j ∈ [ k + 2], uniformly at random from χ and set C 1 ( v x ) = χ j . In a similar wa y , independently gener ate random colo r ings C 2 , . . . , C b , where b = ⌈ 2 e log n ⌉ . F or every ( v x , v y ) / ∈ E ( G ) and v x < D v y , since G is k -degenera te we ha ve | T xy | = t ≤ k + 2. P r [ T xy is fav o r ably c o lored in C i ] = ( k +2)( k +1) t − 1 ( k +2) t − 1 =  k +1 k +2  t − 1 ≥  k +1 k +2  k +1 . The r efore, P r [ T xy is not fav o rably colored in C i ] ≤ 1 −  k +1 k +2  k +1 ≤ e − ( k +1 k +2 ) k +1 . Now taking b = ⌈ 2 e lo g n ⌉ , P r [ [ x,y :( v x < D v y ) , (( v x ,v y ) / ∈ E ( G )) b \ i =1 ( T xy is not fav o rably colored in C i )] ≤ n 2 e − b ( k +1 k +2 ) k +1 < 1 . Hence, P r [ C 1 , . . . , C b satisfy the condition of Lemma 3] > 0 . There fo re, ther e exists a colo ring C 1 , . . . C b , w ith b = ⌈ 2 e log n ⌉ , o f V ( G ) using c o lors fro m the se t { χ 1 , . . . , χ k +2 } s uc h that the co ndition of Lemma 3 is satisfied. Hence b y Lemma 3, cub ( G ) ≤ ( k + 2) · ⌈ 2 e log n ⌉ . Corollary 1. Le t G b e a k -de gener ate gr aph with n vertic es and G ′ a gr aph c onstructe d fr om G with V ( G ′ ) = V ( G ) ∪ V ′ , E ( G ′ ) = E ( G ) ∪ { ( u, v ) | u ∈ V ′ , v ∈ V ( G ′ ) } , and V ( G ) ∩ V ′ = ∅ . Then, cub ( G ′ ) ≤ ( k + 2 ) · ⌈ 2 e lo g n ⌉ . Pr o of. F rom Theorem 1, we know that there exist ( k + 2) · ⌈ 2 e log n ⌉ unit interv a l graphs I i,j , where i ∈ [ k + 2 ], j ∈ [ ⌈ 2 e log n ⌉ ], such that G = T i T j I i,j . Let f i,j be the unit interv al r e presen ta tion of each I i,j as p er the construc tio n in Lemma 3. W e now constr uc t ( k + 2) · ⌈ 2 e log n ⌉ unit interv al gra phs I ′ i,j , wher e i ∈ [ k + 2 ], j ∈ [ ⌈ 2 e log n ⌉ ], s uc h that G ′ = T i T j I ′ i,j . Let f ′ i,j be a unit interv al representation of I ′ i,j . Then for each v ∈ V ( G ′ ), f ′ i,j ( v ) = f i,j ( v ), if v / ∈ V ′ f ′ i,j ( v ) = [ n, 2 n ], if v ∈ V ′ F rom Remark 1 in Lemma 3, every v ∈ V ′ is adjacent with every o ther v er tex in each I ′ i,j since f ′ i,j ( v ) = [ n, 2 n ] , ∀ v ∈ V ′ . 7 Tigh tnes s of Theorem 1 Reca ll that, a r ealizer of a pa rtially ordered set (po set) P is a set of linear ex tensions of P , say R , which satisfy the following condition: for any tw o distinct elements x and y , x ≤ y in P if and only if x ≤ y in L , ∀ L ∈ R . The p oset dimensio n o f P , denoted by dim ( P ), is the minim um integer k such that there exis ts a realize r of P o f cardinality k . Let G P denote the underlying comparability gra ph of P . The n by Theor em 1 in [1], box ( G P ) ≥ dim ( P ) 2 . Let P ( n, p ) b e the pro ba bilit y space of height-2 p osets with n minimal el- ement s forming set A a nd n maximal e lemen ts forming set B , wher e for any a ∈ A and b ∈ B , P r [ a < b ] = p . Erd˝ os, Kierstea d, a nd T rotter in [11] pr o ved that when p = 1 log n , for almo st all po sets P ∈ P ( n, p ), ∆ ( G P ) < δ 1 n log n and dim ( P ) > δ 2 n , whe r e δ 1 and δ 2 are s ome p o sitiv e constants. T he n by Theor em 1 in [1], cub ( G P ) ≥ box ( G P ) ≥ dim ( P ) 2 ≥ δ 2 n 2 . W e know that G P is ∆ ( G P )-degenerate. By Theorem 1, cub ( G P ) ≤ ( ∆ ( G P )+ 2) · ⌈ 2 e log n ⌉ ≤ ( δ 1 n log n + 2) · ⌈ 2 e lo g n ⌉ ≤ cn , where c is some constant. Hence the upper b ound for cubicity given in Theorem 1 is tigh t. 4.2 Deterministi c Algo rithm CONSTRU C T CUB REP( G ) is a deterministic a lgorithm which tak e s a s imple, finite k -degenerate gr aph G a s input and o utputs a cube representation in 8 k α dimensional space i.e., 8 k α unit interv al gra phs I 1 , 1 , . . . , I 1 , 8 k , . . . , I α, 1 , . . . , I α, 8 k such that G = T α i =1 T 8 k j =1 I i,j . In order to ac hieve this, C O NSTR UCT CUB REP ( G ) in vokes the pro cedure CONSTR UCT COL O RING i.e., Algorithm 4.2 (for a detailed version of this pro cedure , see Algorithm 4.4), α times and thereby generates α co lorings C 1 , . . . , C α , where ea c h colo ring uses co lors fro m the se t { χ 1 , . . . , χ 8 k } . Then from each color ing C i , it cons tructs 8 k unit in ter v al g raphs I i, 1 , . . . , I i, 8 k using the construction describ ed in Lemma 3 , whic h is implemen ted in pro cedure CONSTRU CT UNIT INTER V AL GRAPHS. Note that in order fo r G to b e equal to T α i =1 T 8 k j =1 I i,j , Lemma 3 require s that the color ings C 1 , . . . , C α satisfy the following prop erty: for every ( v x , v y ) / ∈ E ( G ), where v x < D v y , ther e exists an i ∈ [ α ] such that the s trong supp ort set T xy of this non-edge is fa vorably colo red in C i . The coloring s C 1 , . . . , C α are generated one by one k eeping this o b jective in mind. At the stag e when we hav e just generated the ( i − 1)-th co loring C i − 1 , if a no n- edge ( v x , v y ) is such tha t its strong supp ort set T xy is already fav o rably co lored in some C j , where j < i , then we say that the non-edge ( v x , v y ) is a lready DONE. Natura lly at each sta ge we hav e to keep tra c k o f the non- e dg es that a r e not yet DONE. In order to do this, we introduce tw o data structures B N N i and F N N i , for a ll i ∈ [ α ] 1 . F or each 1 B N N - Backw ard Non-Neighbor, F N N - F orw ard N on-Neigh b or 8 Algorithm 4.1 CO NSTR UCT CUB REP(G) for y = n to 1 do 1. Initialize B N N 1 [ v y ] ← { v x ∈ V ( G ) | v x < D v y , ( v x , v y ) / ∈ E ( G ) } . 2. Initialize F N N 1 [ v y ] ← { v z ∈ V ( G ) | v y < D v z , ( v y , v z ) / ∈ E ( G ) } . end for 3. SET FLAG ← TRUE. 4. SET i ← 0. while FLAG = TRUE do 5. i++. 6. C i = CONSTRUCT COLORING( i ). for y = 1 to n do 7. SET B N N i +1 [ v y ] ← B N N i [ v y ] \ W ( v y , C i ) 8. SET F N N i +1 [ v y ] ← F N N i [ v y ] \ Y ( v y , C i ) end for 9. If F N N i +1 [ v y ] = ∅ , ∀ v y ∈ V ( G ), then FLAG = F ALSE. end while 10. SET α ← i 11. CONSTRUCT UNIT INTER V A L GRAPHS () v y ∈ V ( G ), B N N i [ v y ] = { v x ∈ V ( G ) | v x is a backward non-neighbor of v y , and ( v x , v y ) is not yet DONE with resp ect to C 1 , . . . , C i − 1 . } F N N i [ v y ] = { v z ∈ V ( G ) | v z is a for ward no n-neigh b or of v y , and ( v y , v z ) is not yet DONE with resp ect to C 1 , . . . , C i − 1 . } It is easy to see that, S v y ∈ V ( G ) B N N i [ v y ] = S v y ∈ V ( G ) F N N i [ v y ] a nd ther efore,  S v y ∈ V ( G ) B N N i [ v y ] = ∅  ⇐ ⇒  S v y ∈ V ( G ) F N N i [ v y ] = ∅  . In Theorem 2 , we show that if we select α to b e at leas t ( ⌈ 2 . 42 log n ⌉ + 1 ), then F N N α +1 [ v y ] = ∅ , ∀ v y ∈ V ( G ). This clearly would mean that all non-edg e s a re DO NE with respe c t to C 1 , . . . , C α . In other words, the condition of Lemma 3 will b e satisfied for C 1 , . . . , C α . The only thing that remains to b e discussed now is how o ur coloring strategy (i.e. the pr ocedure CONSTRUCT COLORING) achiev es the a bov e o b jective, namely B N N α +1 [ v y ] = ∅ a nd F N N α +1 [ v y ] = ∅ , ∀ v y ∈ V ( G ), if α ≥ ( ⌈ 2 . 42 log n ⌉ + 1). T o start with B N N 1 [ v y ] (res pectively F N N 1 [ v y ]) contains all the backw ard (res p ec- tively forward) non-neighbors of v y . The pro cedure CO NSTR UCT COLORING ( i ) generates the i -th co loring C i as follows. It co lors vertices in the re verse de- generacy order star ting from vertex v n . The partial color ing at the sta ge when we hav e colored the vertices v n to v z is denoted by C v z i . Note that C v 1 i = C i . Consider the stage at which the algorithm has already co lored the vertices fro m v n up to v y +1 and is a bout to color v y . That is, we hav e the partial colo r ing C v y +1 i and are ab out to extend it to the partial colo ring C v y i by assigning o ne of 9 Algorithm 4.2 CO NSTR UCT COLORING( i ) /* a bridged * / /*F or a detailed version of this pro cedure, see Algorithm 4.4. All data struct ures are assumed to b e global. Notational Note: Let C v z i denote t h e partial coloring at th e stage when w e ha ve colored the vertices v n to v z . Let C v z = χ c i denote the p artial coloring that results if we exten d C v z +1 i by assi gnin g color χ c to v z .*/ for y = n to 1 do for eac h χ c ∈ { χ 1 , . . . , χ 8 k } do 1. Compute | X ( v y , C v y = χ c i ) | , | Y ( v y , C v y = χ c i ) | , and | Z ( v y , C v y = χ c i ) | as p er equa- tions (2),(3), and (4) resp ectivel y . if | X ( v y , C v y = χ c i ) | ≥ 3 4 | B N N i [ v y ] | and | Y ( v y , C v y = χ c i ) | ≥ 3 4 | Z ( v y , C v y = χ c i ) | then 2. SET C v y i ← C v y = χ c i (i.e. SET C i ( v y ) ← χ c ). 3. SET Y ( v y , C v y i ) ← Y ( v y , C v y = χ c i ) 4. BREAK. end if end for end for for y = 1 to n do 5. Compute W ( v y , C i ) as p er equation (1) 6. SET Y ( v y , C i ) ← Y ( v y , C v 1 i ) end for 7. Return C i . the 8 k pos s ible color s to vertex v y . Let C v y = χ c i denote the partial coloring that results if we extend C v y +1 i by assig ning color χ c to v y . The coloring C i and the partial coloring s C v z i , ∀ v z ∈ V ( G ) and C v z = χ c i , ∀ v z ∈ V ( G ) , χ c ∈ { χ 1 , . . . , χ 8 k } , will be generically called the c olorings asso ciate d with the i -t h stage ( i.e. the i -th inv o cation of CONSTRUCT COLORING). With resp ect to colo rings C 1 , . . . , C i − 1 and some c oloring C ′ i asso ciated with the i -th stage , we define the following sets: W ( v w , C ′ i ) = { v x ∈ B N N i [ v w ] | the strong supp ort set T xw of non-edg e (1) ( v x , v w ) is favorably co lored in C ′ i } X ( v w , C ′ i ) = { v x ∈ B N N i [ v w ] | the weak supp ort set S xw of non-edge (2) ( v x , v w ) is favorably co lored in C ′ i } Y ( v w , C ′ i ) = { v z ∈ F N N i [ v w ] | the stro ng supp ort set T wz of non-edge (3) ( v w , v z ) is fav or ably color ed in C ′ i } Z ( v w , C ′ i ) = { v z ∈ F N N i [ v w ] | the weak supp o rt set S wz of non-edg e (4) ( v w , v z ) is fav or ably color ed in C ′ i } Naturally , we want to give a color χ c to v y such that a lar ge num b er of (not yet DONE) no n-edges incident on v y get DONE. With resp ect to the coloring s C 1 , . . . , C i − 1 and the partial coloring C v y = χ c i , we define the status of a no n-edge 10 Algorithm 4.3 CO NSTR UCT UNIT INTER V AL GRAPHS() /*All data structures are assumed to b e global. */ 1. INITIALI ZE l ( f i,j ( v y )) ← 0 , r ( f i,j ( v y )) ← n, ∀ y ∈ [ n ] , i ∈ α, j ∈ [8 k ] for i = 1 to α do for y = n to 1 do 2. SET j ← c , such t hat C i ( v y ) = χ c 3. SET l ( f i,j ( v y )) ← y + n 4. SET r ( f i,j ( v y )) ← y + 2 n for eac h v ∈ N b G ( v y ) do if ( C i ( v ) 6 = j ) ∩ ( l ( f i,j ( v )) = 0) then 5. SET l ( f i,j ( v )) ← y 6. SET r ( f i,j ( v )) ← y + n end if end for end for end for 7. Output f i,j ( v y ) , ∀ y ∈ [ n ] , i ∈ α, j ∈ [8 k ] incident on v y as follows: A non-edg e ( v y , v z ) ∈ F N N i [ v y ] is DONE 2 if T y z is fav orably colo red in C v y = χ c i and is NOT- DONE if T y z is not fav or a bly co lored in C v y = χ c i . A non-e dge ( v x , v y ) ∈ B N N i [ v y ] is HO P ELESS 3 if S xy (whic h happ ens to b e a prop er subset of T xy ) is not fav ora bly colored in C v y = χ c i and is HOPEFUL if S xy is fav ora bly colo red in C v y = χ c i . So when we de c ide a color for v y , o ur int ention is to make a large fractio n of the HOPEFUL non-edg es of F N N i [ v y ] (i.e. the s et Z ( v y , C v y = χ c i )), DONE and to make a lar ge fractio n of B N N i [ v y ], HOPEFUL. More formally , we want the algorithm to a ssign a color χ c to v y such that the following tw o conditions a re satisfied. (i) | X ( v y , C v y = χ c i ) | ≥ 3 4 | B N N i [ v y ] | , and (ii) | Y ( v y , C v y = χ c i ) | ≥ 3 4 | Z ( v y , C v y = χ c i ) | . The obvious question then is whether such a colo r χ c alwa ys exists, for e ac h v y ∈ V ( G ). Lemma 4 answers this question in the affirmative. It follows that, the nu mber of non-edges tha t a r e not yet DONE with resp ect to colo rings C 1 , . . . C i is at most a co nstan t fractio n of the num b er of non- e dges that were not DONE with r espect to coloring s C 1 , . . . C i − 1 . This is forma lly proved in Lemma 5. Tha t B N N α +1 [ v y ] = ∅ and F N N α +1 [ v y ] = ∅ , ∀ v y ∈ V ( G ), is a cons equence of this and is forma lly proved in Theorem 2 . Lemma 4. F or every i ∈ [ α ] , v y ∈ V ( G ) , (i) | X ( v y , C i ) | ≥ 3 4 | B N N i [ v y ] | , and (ii) | Y ( v y , C i ) | ≥ 3 4 | Z ( v y , C i ) | . 2 Recall that we had defined earlier that a non- edge ( v x , v y ) is D O NE with resp ect to a list of colorings C 1 , . . . , C i − 1 if T xy w as fav orably colored in some C j , where j < i . Here we ext en d this notion, by allo wing the p artial coloring C v y = χ c i also in th e list. 3 A HOPELESS non-edge ( v x , v y ) will not b e DO N E with respect to C 1 , . . . , C i if we set C i ( v y ) = χ c , irresp ectiv e of the color given to v y − 1 , . . . , v 1 . 11 Pr o of. The statement of the lemma is obvious if the BREAK sta tement in Step 4 of CONSTR UCT COLO RING( i ) (abridg ed version) is ex ecuted, for every i ∈ [ α ] and v y ∈ V ( G ). In order to prove that the BREAK statement will b e e xecuted, it is sufficient to show that there exis ts a co lo r χ c ∈ { χ 1 , . . . , χ 8 k } such that | X ( v y , C v y = χ c i ) | ≥ 3 4 | B N N i [ v y ] | and | Y ( v y , C v y = χ c i ) | ≥ 3 4 | Z ( v y , C v y = χ c i ) | . Since the vertices in Z ( v y , C v y = χ c i ) or Z ( v y , C i ) do not dep end on the colo rs given to v 1 , . . . v y , w e hav e Z ( v y , C v y = χ c i ) = Z ( v y , C i ) . Hence, Z ( v y , C v y = χ c i ) and Z ( v y , C i ) can b e us ed interchangeably . Let A = B N N i [ v y ] × Z ( v y , C i ). Let < v x , v z > b e an e lemen t o f A . W e say a colo r χ c is go o d for < v x , v z > , if v x ∈ X ( v y , C v y = χ c i ) a nd v z ∈ Y ( v y , C v y = χ c i ). In other words, χ c is go o d for < v x , v z > , if bo th S xy and T y z are fav o rably colored in C v y = χ c i . S xy is fav o rably colo red in C v y = χ c i , if χ c / ∈ P , where P = {C v y = χ c i ( v w ) | v w ∈ N f G ( v x ) , v y < D v w } . Since | N f G ( v x ) | ≤ k , | P | ≤ k . Therefor e, there ar e at least 8 k − k = 7 k po ssible v alues that χ c can take s uc h tha t S xy is fav orably colo red in C v y = χ c i . F or T y z also to be fa vorably color ed in C v y = χ c i , the only thing r equired is that χ c 6 = C v y = χ c i ( v z ), since v z ∈ Z ( v y , C i ) and therefore S y z is already fav or ably color ed. This implies tha t there a re at least 7 k − 1 po ssible v alues tha t χ c can tak e s uc h that b oth S xy and T y z are favorably colored in C v y = χ c i . In other w or ds, there are at least 7 k − 1 go o d co lors for < v x , v z > . Thu s for ea c h element in A , there ar e at least 7 k − 1 color s go o d for it. F or ea c h color χ j ∈ { χ 1 , . . . , χ 8 k } , let S j = { < v x , v z > ∈ A | χ j is go o d fo r < v x , v z > } = X ( v y , C v y = χ j i ) × Y ( v y , C v y = χ j i ). Since there a re at least (7 k − 1) colors go o d for each element in A , Σ j ∈ [8 k ] | S j | ≥ (7 k − 1) | A | . Then b y pigeonhole principle, there exists a c ∈ [8 k ] such that | S c | = | X ( v y , C v y = χ c i ) | · | Y ( v y , C v y = χ c i ) | ≥ (7 k − 1) 8 k | A | = 7 k − 1 8 k | B N N i [ v y ] | · | Z ( v y , C i ) | ≥ 3 4 | B N N i [ v y ] | · | Z ( v y , C i ) | elements o f A . In other words, | X ( v y , C v y = χ c i ) | ≥ 3 4 | B N N i [ v y ] | and | Y ( v y , C v y = χ c i ) | ≥ 3 4 | Z ( v y , C v y = χ c i ) | . ⊓ ⊔ Lemma 5. L et m i = Σ y ∈ [ n ] | F N N i [ v y ] | . Then m i +1 ≤ 7 16 m i . Pr o of. F rom Step 8 of CONSTRUCT CUB REP( G ), we have | F N N i +1 [ v y ] | = | F N N i [ v y ] | − | Y ( v y , C i ) | ≤ | F N N i [ v y ] | − 3 4 | Z ( v y , C i ) | (using Lemma 4). T ak- ing summation ov er all y ∈ [ n ], we get m i +1 ≤ m i − 3 4 Σ y ∈ [ n ] | Z ( v y , C i ) | = m i − 3 4 Σ y ∈ [ n ] | X ( v y , C i ) | . The la st equality c o mes fro m the fact that b oth Σ y ∈ [ n ] | X ( v y , C i ) | and Σ y ∈ [ n ] | Z ( v y , C i ) | r e presen t the num b er of HOP EFUL non-edges in G with r espect to colorings C 1 , . . . , C i . F rom Lemma 4, we hav e | X ( v y , C i ) | ≥ 3 4 | B N N i [ v y ] | . Ther e fore, m i +1 ≤ m i − ( 3 4 ) 2 Σ y ∈ [ n ] | B N N i [ v y ] | . Since Σ y ∈ [ n ] | B N N i [ v y ] | = Σ y ∈ [ n ] | F N N i [ v y ] | , we get m i +1 ≤ m i − ( 3 4 ) 2 Σ y ∈ [ n ] | F N N i [ v y ] | = m i − 9 16 m i = 7 16 m i . ⊓ ⊔ Theorem 2. L et G b e a k -de gener ate gr aph. A lgorithm CONSTRUCT CUB REP( G ) c onst r u cts a valid 8 k ( ⌈ 2 . 4 2 log n ⌉ + 1) dimensional cub e r epr esentation for G . Pr o of. The a lgorithm constructs α coloring s C 1 , C 2 , . . . , C α of V ( G ), where each coloring uses co lors from the set { χ 1 , . . . , χ 8 k } . F r o m Lemma 5 , we hav e m i +1 ≤ 12 7 16 m i . Also , m 1 = | Σ y ∈ [ n ] F N N 1 [ v y ] | ≤ n 2 . Putting α = ( ⌈ 2 . 4 2 log n ⌉ + 1 ), we get m α ≤ 1. That is, for e very y ∈ [ n ], F N N α +1 [ v y ] = E M P T Y . This means that, for e v ery ( v x , v y ) / ∈ E ( G ), where v x < D v y , there exists a n i ∈ [ α ] such that T xy is fav o r ably color ed in C i . Then b y Lemma 3 , cub ( G ) ≤ 8 k ( ⌈ 2 . 42 log n ⌉ + 1). The procedur e CONSTR UCT UNIT INTER V AL GRAPHS constructs 8 k ( ⌈ 2 . 4 2 log n ⌉ + 1) unit interv al gr a phs whose in terse c tio n g iv es G , as describ ed in Lemma 3 . Thus we prove the theorem. Running Time Analysis Lemma 6. The pr o c e dur e CONSTRUCT COLORING( i ) c an b e implemente d to run in O ( k m i + kn ) t ime, wher e m i = Σ y ∈ [ n ] | F N N i [ v y ] | . Pr o of. A detailed description of the pro cedure is given in Alg o rithm 4.4. T o implemen t the pro cedure efficiently , we make use of a n ( n × 8 k ) 0-1 matrix, hereafter called F N C (F orward Neighbor Co lo r), and tw o ( n × n ) 0-1 matrices named H OP E M AT R I X and D ON E M AT R I X r e spectively . A t the b egin- ning of the pro cedure each of these ma trices hav e all entries set to 0. As the pro cedure progr esses, we change some of the en tr ies to 1 in such a way that, ∀ w ∈ [ n ] , j ∈ [8 k ] , F N C [ w ][ j ] = 1 ⇐ ⇒ ∃ v z ∈ N f G ( v w ) such that v z is already colored by the pro cedure with co lor χ j . ∀ w, z ∈ [ n ] , v w ∈ B N N i [ v z ] , H O P E M AT R I X [ w ][ z ] = 1 ⇐ ⇒ S wz is already fav orably colored by the pro cedure. ∀ w, z ∈ [ n ] , v w ∈ B N N i [ v z ] , D ON E M AT RI X [ w ][ z ] = 1 ⇐ ⇒ T wz is already fav orably colored by the pro cedure. In o rder for the ab o ve matrices to satisfy their r espective pr operties , the only thing that needs to b e done is to upda te these matr ic es at each stage of the pro cedure. Consider the stage at which the pro cedure is e xtending par - tial coloring C v y +1 i to C v y i by assigning colo r χ c to v y . At this sta ge, the ma- trices F N C , H OP E M AT R I X and DO N E M AT RI X are updated as de- scrib ed in steps 11 ( a ), 1 2( a ) and 13( a ) resp ectively . Note that this ca n b e done in O ( | B N N i [ v y ] | + | F N N i [ v y ] | + | N b G ( v y ) | ) time. Steps 4(a)-(b), 5(a)-(b) a nd 6(a)-(b) compute X ( v y , C v y = χ c i ) , Y ( v y , C v y = χ c i ) and Z ( v y , C v y = χ c i ) r e s pectively in O ( | B N N i [ v y ] | + | F N N i [ v y ] | ) time. Computing W ( v y , C i ) is done in step 15 (a)-(b) in O ( | B N N i [ v y ] | ) time. Since steps 4 to 14, in the worst cas e , a re r un for each v y ∈ V ( G ) , χ c ∈ { χ 1 , . . . , χ 8 k } , the pro cedure runs in O ( k ( Σ y ∈ [ n ] ( | B N N i [ v y ] | + | F N N i [ v y ] | ) + Σ y ∈ [ n ] | N b G ( v y ) | )) time. W e know that Σ y ∈ [ n ] ( | B N N i [ v y ] | + | F N N i [ v y ] | ) = 2 m i and Σ y ∈ [ n ] | N b G ( v y ) | = m ≤ k n . Hence the Lemma. ⊓ ⊔ Theorem 3. CONSTRUCT CUB REP( G ) runs in O ( n 2 k ) time. Pr o of. The a lgorithm inv okes the function CONSTRUCT COLORING( i ) α times to construct color ings C 1 , . . . , C α of V ( G ). By Lemma 6, to construct these α co lorings it requires O ( Σ α i =1 ( m i k ) + αk n ) time. F ro m Lemma 5 , we get that Σ α i =1 ( m i ) is O ( m ). Since α = ( ⌈ 2 . 42 log n ⌉ + 1 ), the running time of 13 Algorithm 4.4 CO NSTR UCT COLORING( i ) /* deta iled * / /*All data structures are assumed to b e global. Notational Note: Let C v z i denote t h e partial coloring at th e stage when w e ha ve colored the vertices v n to v z . Let C v z = χ c i denote the p artial coloring that results if we exten d C v z +1 i by assi gnin g color χ c to v z . */ 1. Initialize F N C [ w ][ j ] ← 0 , ∀ w ∈ [ n ] , j ∈ [8 k ] 2. Initialize H O P E M AT RI X [ w ][ z ] ← 0 , ∀ w , z ∈ [ n ] 3. Initialize D ON E M AT RI X [ w ][ z ] ← 0 , ∀ w, z ∈ [ n ] for y = n to 1 do for eac h χ c ∈ { χ 1 , . . . , χ 8 k } do 4. Compute X ( v y , C v y = χ c i ) /*as d escribed in steps (a) and (b) b elo w */ (a) Initialize X ( v y , C v y = χ c i ) ← ∅ (b) ∀ v x ∈ B N N i [ v y ], if F N C [ x ][ c ] = 0, th en SET X ( v y , C v y = χ c i ) ← X ( v y , C v y = χ c i ) ∪ { v x } 5. Compute Y ( v y , C v y = χ c i ) /*as d escribed in steps (a) and (b) b elo w */ (a) Initialize Y ( v y , C v y = χ c i ) ← ∅ (b) ∀ v z ∈ F N N i [ v y ], if ( H O P E M AT RI X [ y ][ z ] = 1) and  C v y = χ c i ( v z ) 6 = χ c  , then SET Y ( v y , C v y = χ c i ) ← Y ( v y , C v y = χ c i ) ∪ { v z } 6. Compute Z ( v y , C v y = χ c i ) /*as d escribed in steps (a) and (b) b elo w */ (a) Initialize Z ( v y , C v y = χ c i ) ← ∅ (b) ∀ v z ∈ F N N i [ v y ], if H O P E M AT RI X [ y ][ z ] = 1, then SET Z ( v y , C v y = χ c i ) ← Z ( v y , C v y = χ c i ) ∪ { v z } if | X ( v y , C v y = χ c i ) | ≥ 3 4 | B N N i [ v y ] | and | Y ( v y , C v y = χ c i ) | ≥ 3 4 | Z ( v y , C v y = χ c i ) | then 7. SET C v y i ← C v y = χ c i (i.e. SET C i ( v y ) ← χ c ). 8. SET X ( v y , C v y i ) ← X ( v y , C v y = χ c i ) 9. SET Y ( v y , C v y i ) ← Y ( v y , C v y = χ c i ) 10. SET Z ( v y , C v y i ) ← Z ( v y , C v y = χ c i ) 11. Up date F N C matrix. /* as describ ed in step (a) b elow */ (a) ∀ v x ∈ N b G ( v y ), S ET F N C [ x ][ c ] ← 1 12. Up date H O P E M AT RI X /* as describ ed in step (a) b elo w */ (a) ∀ v x ∈ X ( v y , C v y i ), SET H O P E M AT RI X [ x ][ y ] ← 1 13. Up date D ON E M AT RI X /* as describ ed in step (a) b elo w */ (a) ∀ v z ∈ Y ( v y , C v y i ), SET D ON E M AT RI X [ y ][ z ] ← 1 14. BREAK. end if end for end for for y = 1 to n do 15. Compute W ( v y , C i ) /*as d escribed in steps (a) and (b) b elo w */ (a) Initialize W ( v y , C i ) ← ∅ (b) ∀ v x ∈ B N N i [ v y ], if D ON E M AT RI X [ x ][ y ] = 1, then SET W ( v y , C i ) ← W ( v y , C i ) ∪ { v x } 16. SET Y ( v y , C i ) ← Y ( v y , C v 1 i ) end for 17. Return C i . 14 the while lo op in CO NSTR UCT CUB REP( G ) is O ( mk + n k lo g n ). It is eas y to see tha t the pro cedure CONSTRUCT UNIT INTER V AL GRAPHS() runs in O ( nk log n ) time. Since m ≤ n 2 , CONSTRUCT CUB REP( G ) runs in O ( n 2 k ) time. 5 Bo xicity , Cubicit y , and Cr ossing Number Crossing num b er of a gr aph G , denoted as C R ( G ), is the minimum n umber of crossing pairs o f edges , ov er all drawings of G in the pla ne. A graph G is planar if and o nly if C R ( G ) = 0. Determination of the crossing num b er is an NP-c o mplete problem. The following theorem is due to Pach and T´ oth [18 ] Theorem 4. F or a gr aph G with n vertic es and m ≥ 7 . 5 n e dges, C R ( G ) ≥ 1 33 . 75 m 3 n 2 , and this estimate is tight up to a c onst ant factor. The following claims follow from the ab ov e theorem. Claim 1. F or a g raph G on n v ertices and m edges, if C R ( G ) ≤ t , then d av ( G ) ≤ 2( 33 . 75 t n ) 1 / 3 + 15. Pr o of. If m < 7 . 5 n , then d av ( G ) < 15. Otherwise, we have m ≤ (3 3 . 75 n 2 t ) 1 / 3 implying that d av ( G ) ≤ 2( 33 . 75 t n ) 1 / 3 . ⊓ ⊔ Claim 2. F or a graph G on n vertices and m edges, if C R ( G ) = t , then G is  6 . 5 t 1 / 4 + 15  -degenera te. Pr o of. F rom the definition o f c r ossing num b er we know that C R ( G ) ≤  m 2  ≤ n 4 . Hence, n ≥ t 1 / 4 . Then b y Cla im 1, d av ( G ) ≤ 6 . 5 t 1 / 4 + 15 . Thus G is  6 . 5 t 1 / 4 + 15  -degenera te. ⊓ ⊔ Lemma 7. Consider a gr aph G whose vertic es ar e p artitione d into two p arts namely V A and V B . Th at is, V ( G ) = V A ⊎ V B . Then, box ( C B ( G )) ≤ 2 box ( S B ( G )) . Pr o of. Pro of of this lemma is v e r y similar to the pro of of Lemma 3 in [7] and hence we only give a brief outline o f it here. Ass ume b ox ( S B ( G )) = r . Then by Lemma 1 , there exist r interv al gra phs I 1 , . . . , I r such that S B ( G ) = I 1 ∩ I 2 ∩ · · · ∩ I r . F or each i ∈ [ r ], let f i denote an interv al r epresentation o f I i . F r om these r int er v al g raphs we construct 2 r interv a l gr aphs I ′ 1 , . . . , I ′ r , I ′′ 1 , . . . , I ′′ r as outlined below. Let f ′ i , f ′′ i denote interv al repre s en tations of I ′ i and I ′′ i resp ectiv e ly , where i ∈ [ r ]. Construction of f ′ i : ∀ u ∈ V A , f ′ i ( u ) = f i ( u ) . ∀ u ∈ V B , f ′ i ( u ) = [ min v ∈ V B ( l ( f i ( v ))) , r ( f i ( u ))] . Construction of f ′′ i : ∀ u ∈ A, f ′′ i ( u ) = f i ( u ) . ∀ u ∈ B , f ′′ i ( u ) = [ l ( f i ( u )) , max v ∈ V B ( r ( f i ( v )))] . 15 W e leav e it to the reader to verify tha t C B ( G ) = T r i =1 ( I ′ i ∩ I ′′ i ). ⊓ ⊔ Lemma 8. Consider a gr aph G . L et vertic es of G b e p artitione d into two p arts namely V A and V B . That is, V ( G ) = V A ⊎ V B . Then, box ( G ) ≤ 2 box ( S B ( G )) + box ( G B ) . Pr o of. Let G ′ be the graph with V ( G ′ ) = V ( G ) and E ( G ′ ) = E ( G ) ∪ { ( u, v ) | u ∈ V A , v ∈ V ( G ′ ) } . That is, each u ∈ V A is made a universal vertex in G ′ . Obser v e that G = C B ( G ) ∩ G ′ . Then by Lemma 1, we hav e box ( G ) ≤ box ( C B ( G )) + box ( G ′ ). Applying Lemma 7 , we get box ( G ) ≤ 2 box ( S B ( G )) + box ( G ′ ) (5) Claim 3. box ( G ′ ) ≤ box ( G B ). Clearly , G ′ is obtained fr o m G B by adding univ ers al v ertices o ne after the other. Since adding a universal vertex to a gra ph does not increa se its b oxic- it y , box ( G ′ ) ≤ box ( G B ). Combining Inequalit y 5 and Claim 3, w e get b ox ( G ) ≤ 2 box ( S B ( G )) + box ( G B ). ⊓ ⊔ 5.1 Bo xicity and Crossing N um b er Theorem 5. F or a gr aph G with C R ( G ) = t , box ( G ) ≤ 66 · t 1 4 ⌈ log 4 t ⌉ 3 4 + 6 . Pr o of. Consider a dr a wing P of G with t crossings. W e s ay a v ertex v p articip ates in a given cross ing in P , if at least one of the edges of the giv en crossing is incident on v . Partition the vertices of G into t wo parts, namely V A and V B , such that V B = { v ∈ V ( G ) | v participates in some cros s ing in P } and V A = V ( G ) \ V B . Then by Lemma 8, box ( G ) ≤ 2 box ( S B ( G )) + box ( G B ) . Observe that S B ( G ) is a planar g raph and hence its b oxicit y is at most 3 (see [21]). Therefore, box ( G ) ≤ 6 + box ( G B ). F or ease of notation, let H ≡ G B . Then, box ( G ) ≤ 6 + box ( H ) . (6) W e hav e C R ( H ) = C R ( G ) = t . Let n = | V ( H ) | and m = | E ( H ) | . A t mos t 4 vertices par ticipate in a g iv en cros sing. Since each vertex in H participa tes in some cross ing in P , we get n ≤ 4 t. Let V ( H ) = { v 1 , . . . , v n } . Let v 1 , . . . , v n be a n order ing of the vertices of H , such that for ea c h i ∈ [ n ], d H i ( v i ) ≤ d H i ( v ) , ∀ v ∈ V ( H i ), where H i denotes the subgraph of H induced o n vertex se t { v i , . . . , v n } . Let k =  33 . 75 3  1 4  t ⌈ log 4 t ⌉  1 4 . 16 Let x = min( { i ∈ [ n ] | d H i ( v i ) > k } ). Partition V ( H ) in to tw o parts, namely V C = { v 1 , . . . , v x − 1 } and V D = { v x , . . . , v n } . Then by Lemma 8, box ( H ) ≤ 2 box ( S D ( H )) + box ( H D ) . Note that S D ( H ) is k -degenerate. If k = 1 , then S D ( H ) is a forest and hence its b oxicity is at most 2. Supp ose k > 1. Then by Theo rem 1 , b ox ( S D ( H )) ≤ cub ( S D ( H )) ≤ ( k + 2) ⌈ 2 e log n ⌉ ≤ 1 2 k ⌈ log(4 t ) ⌉ ≤ 12  33 . 75 3  1 4 t 1 4 ⌈ log 4 t ⌉ 3 4 . Thus we hav e, box ( H ) ≤ 24  33 . 75 3  1 4 t 1 4 ⌈ log 4 t ⌉ 3 4 + box ( H D ) . (7) Since H D ≡ H x , v x is a minimum degree vertex of H D . Therefo r e, d av ( H D ) > d H D ( v x ) > k . Then by Claim 1, we have k =  33 . 75 3  1 4  t ⌈ log 4 t ⌉  1 4 < d av ( H D ) ≤ 2  33 . 75 t | V ( H D ) |  1 / 3 + 15 . F rom this, we get | V ( H D ) | ≤ 48 3 4 (33 . 75 t ) 1 4 ⌈ log 4 t ⌉ 3 4 . Since boxicity of a g r aph is at most half the num b er o f its vertices[19] , we get box ( H D ) ≤ 48 3 4 (33 . 75 t ) 1 4 ⌈ log 4 t ⌉ 3 4 2 . Substituting this in Inequality 7, we g et box ( H ) ≤ 66 t 1 4 ⌈ log 4 t ⌉ 3 4 Therefore from Ineq ua lit y 6 ,we g e t box ( G ) ≤ 66 t 1 4 ⌈ log 4 t ⌉ 3 4 + 6 . Tigh tnes s of Theorem 5: Let P ( n, p ) b e the pro babilit y spa ce of height-2 po sets with n minimal elements for ming set A and n maximal elements forming set B , where for any a ∈ A a nd b ∈ B , P r [ a < b ] = p . Er d˝ o s, Kierstea d, and T rotter in [11 ] prov ed that when p = 1 log n , for almost a ll po sets P ∈ P ( n, p ), ∆ ( G P ) < δ 1 n log n and dim ( P ) > δ 2 n , where δ 1 and δ 2 are some p ositive cons tan ts. Then by Theor e m 1 in [1 ], box ( G P ) ≥ dim ( P ) 2 ≥ δ 2 n 2 . Let t = C R ( G P ) a nd let m denote the num b er of edg es of G P . It follows from definition o f cr ossing n umber that t ≤  m 2  ≤ m 2 ≤ ( n∆ ( G P )) 2 ≤ ( δ 1 n 2 log n ) 2 ≤ δ 2 1 n 4 (log n ) 2 . Since t ≤ m 2 ≤ n 4 , we hav e n ≥ t 1 / 4 and thereby log n ≥ 1 4 log t . Thus, t ≤ δ 2 1 n 4 ( 1 4 log t ) 2 = 16 δ 2 1 n 4 (log t ) 2 . F rom Theorem 5, w e hav e box ( G P ) ≤ ct 1 / 4 (log t ) 3 / 4 + d ≤ cn (log t ) 1 / 4 + d , where c a nd d are some co nstan ts. Therefor e , the b ound given by Theorem 5 is tight up to a factor of O ((log t ) 1 4 ). 17 5.2 Cubicit y and Crossing Numbe r Theorem 6. F or a gra ph G with C R ( G ) = t , cu b ( G ) ≤ 6 log 2 n +  6 . 5 t 1 / 4 + 17  ⌈ 2 e log(4 t ) ⌉ . Pr o of. Consider a drawing P of G with t crossings. As in Theo rem 5, partition the vertices of G int o t wo parts, namely V A and V B , s uc h that V B = { v ∈ V ( G ) | v participates in so me cro ssing in P } and V A = V ( G ) \ V B . Let G ′ be the g r aph with V ( G ′ ) = V ( G ) and E ( G ′ ) = E ( G ) ∪ { ( u, v ) | u ∈ V A , v ∈ V ( G ′ ) } . That is, e ac h u ∈ V A is made a universal vertex in G ′ . Obser v e that G = C B ( G ) ∩ G ′ . Then by Lemma 2 , cub ( G ) ≤ cub ( C B ( G )) + cu b ( G ′ ) It is shown in [6] that cubicit y of a gra ph is at most log 2 n times its b oxicit y . Applying this r esult, we get cub ( G ) ≤ (log 2 n ) box ( C B ( G )) + cub ( G ′ ) ≤ (2 log 2 n ) box ( S B ( G )) + cub ( G ′ ) (by Lemma 7 ) Observe that S B ( G ) is a planar g raph and hence its b oxicit y is at most 3 (see [21]). Therefor e , cub ( G ) ≤ 6 log 2 n + cu b ( G ′ ) (8) Observe that G ′ is the g raph with V ( G ′ ) = V ( G B ) ⊎ V A and E ( G ′ ) = E ( G B ) ∪ { ( u, v ) | u ∈ V A , v ∈ V ( G ′ ) } . Since C R ( G B ) = C R ( G ) = t , by Claim 2 , G B is  6 . 5 t 1 / 4 + 15  -degenera te. Then b y Corolla ry 1 , cub ( G ′ ) ≤  6 . 5 t 1 / 4 + 17  ⌈ 2 e log( | V B | ) ⌉ . W e know that at most 4 vertices par ticipate in a given crossing . Since each vertex in G B participates in some crossing in P , we get | V B | ≤ 4 t. Thu s , cub ( G ′ ) ≤  6 . 5 t 1 / 4 + 17  ⌈ 2 e log(4 t ) ⌉ . Substituting for cub ( G ′ ) in Inequa l- it y (8), we get cub ( G ) ≤ 6 lo g 2 n +  6 . 5 t 1 / 4 + 17  ⌈ 2 e log(4 t ) ⌉ . 6 Cubicit y of R and om Graphs Given n and m , in orde r to prove tha t a lmo st all gr aphs in G ( n, m ) mo del have cubicit y O ( 2 m n log n ), we first show that cubicit y of almost all gr aphs in G ( n, p ) mo del, where p =  2 m n  1 n − 1 = m ( n 2 ) , is O ( 2 m n log n ). W e then use a re sult in [2] to conv e r t the result for gr aphs in G ( n, p ) mo del to those in G ( n, m ) mo del. T o show that a lmost all graphs in G ( n, p ) mo del hav e cubicity O ( 2 m n log n ), we prove the following lemma . Then by Theorem 1, the de s ired r e sult follows. Lemma 9. F or a r andom gr aph G ∈ G ( n, p ) , wher e p = c n − 1 and 1 ≤ c ≤ n − 1 , P r [ G is 4 ec -de gener ate ] ≥ 1 − 1 Ω ( n 2 ) . 18 Pr o of. In order to show that a given gr a ph G is k -degener ate it is eno ugh to show that every induced subgr a ph o f G has average deg ree at most k . That is, for every H whic h is an induced subgra ph of G , | E ( H ) | ≤ | V ( H ) | k 2 . Below we prov e that for almost all gr a phs G ∈ G ( n, p ), every induced subgraph H of G has E ( H ) < | V ( H ) | 2 ec . W e use the fo llowing version of the Cherno ff b ound (re fer pag e 64 o f [1 7]) in our pro of P r [ X ≥ (1 + δ ) µ ] <  e δ (1 + δ ) (1+ δ )  µ ≤ 1 2 (1+ δ ) µ log 2 ( 1+ δ e ) , where X is a summation of indep endent Bernoulli random v ariables, µ ≥ E [ X ], and δ is any p ositive constant. Let G ∈ G ( n, p ) b e a ra ndom g raph, where p = c n − 1 . Let H be an induced subgraph o f G with | V ( H ) | = nα , where 0 < α ≤ 1. Let Y H be a random v ariable that r epresent s the num b er of edges in H . F or a n y v ∈ V ( H ), let d H ( v ) denote the deg ree of v in H . Then, E [ d H ( v )] = p ( nα − 1) = c ( nα − 1) n − 1 ≤ c ( nα − α ) n − 1 ≤ cα and E [ Y H ] = E [ 1 2 Σ v ∈ V ( H ) d H ( v )] = 1 2 Σ v ∈ V ( H ) E [ d H ( v )] ≤ nα 2 c 2 . Let δ = 4 e α − 1 and µ = nα 2 c 2 ≥ E [ Y H ]. Applying Chernoff bound, we g et P r [ Y H ≥ 2 e nαc ] ≤ 1 2 2 enαc log 2 ( 4 α ) . Here we split the pro of into tw o ca ses: case 1 4 ≤ α ≤ 1 : Then, P r [ Y H ≥ 2 enαc ] ≤ 1 2 (1 / 2) enc log 2 ( 4 1 ) = 1 2 enc . Since c ≥ 1, we get P r [ Y H ≥ 2 enαc ] ≤ 1 2 en . Applying unio n b ound it follows tha t, P r [ [ H : | V ( H ) |≥ n 4 Y H ≥ 2 enαc ] ≤ 1 2 en 1 X α = 1 4  n nα  ≤ 2 n 2 en = 1 2 ( e − 1) n . case 0 < α < 1 4 : Here we use the follo wing expr e s sion g iv en in page 17 of [13] while taking union b ound:  n nα  ≤ 2 nH ( α ) , where H ( α ) = α log 2 ( 1 α ) + (1 − α ) log 2 ( 1 1 − α ) is the bina ry ent r op y function. This inequality can b e proved using the Stirling’s formula for fa ctorials. Since α < 1 4 , we hav e α log 2 ( 1 α ) > (1 − α ) log 2 ( 1 1 − α ). Therefor e , H ( α ) ≤ 2 α log 2 1 α . Hence, P r [ [ 1 ≤ nα ≤ n 4 [ H : | V ( H ) | = nα ( Y H ≥ 2 enαc )] ≤ n 4  n nα  1 2 2 enαc log 2 ( 4 α ) ≤ 2 log 2 ( n 4 ) 2 2 nα log 2 ( 1 α ) 2 2 enαc log 2 ( 4 α ) ≤ 2 2 nα log 2 ( 1 α )+log 2 n 2 4 enαc +2 enαc log 2 ( 1 α ) = 1 2 4 enαc +(2 ec − 2) nα log 2 ( 1 α ) − log 2 n 19 Since c ≥ 1 a nd α ≥ 1 n , we get P r [ [ 1 ≤ nα ≤ n 4 [ H : | V ( H ) | = nα ( Y H ≥ 2 enαc )] ≤ 1 2 4 e +(2 ec − 2) nα log 2 ( 1 α ) − log 2 n (9) It is eas y to see that the function f ( α ) = α log 2 ( 1 α ) is an incr easing function, when α < 1 4 . W e have 1 n ≤ α < 1 4 . Hence f ( α ) ≥ f (1 /n ) = log 2 n n , when α < 1 4 . Applying this to Ineq ualit y 9, we get P r [ [ 1 ≤ nα ≤ n 4 [ H : | V ( H ) | = nα ( Y H ≥ 2 enαc )] ≤ 1 2 4 e +(2 ec − 3) log 2 n Thu s we say that the probability of a n y subgraph o f G to have its av era g e degree grea ter tha n (4 ec + 1) is at most 1 Ω ( n 2 ) . In other words, G is 4 ec -deg enerate with probability a t least 1 − 1 Ω ( n 2 ) . ⊓ ⊔ Theorem 7. F or a r andom gr aph G ∈ G ( n, p ) , wher e p = c n − 1 and 1 ≤ c ≤ n − 1 , P r [ cub ( G ) / ∈ O ( c log n )] ≤ 1 Ω ( n 2 ) . Pr o of. Pro of follows directly fro m The o rem 1 a nd Lemma 9. It is shown in page 35 of [2] that , P m ( Q ) ≤ 3 √ mP p ( Q ) where Q is a proper t y of g raphs of o rder n , and P m ( Q ) and P p ( Q ) are the probabilities of a graph chosen at random from the G ( n, m ) or the G ( n, p ) mode ls resp ectiv e ly to hav e prop erty Q given that p = m ( n 2 ) =  2 m n  1 n − 1 . Note that for any co nnected gra ph G with at lea st 2 v er tices, 2 m n ≥ 2( n − 1) n ≥ 1. Since we ar e only interested in c o nnected graphs, we assume 2 m n ≥ 1. Then by The o rem 7, for a random gr aph G ∈ G ( n, p ), where p =  2 m n  1 n − 1 , P r [ cub ( G ) / ∈ O ( 2 m n log n )] ≤ 1 Ω ( n 2 ) . Co m bining this r esult with the re sult s hown in [2], we say that for a random graph G ∈ G ( n, m ), P r [ cu b ( G ) / ∈ O ( 2 m n log n )] ≤ 3 √ m Ω ( n 2 ) ≤ 1 Ω ( n ) . 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