The complexity of determining the rainbow vertex-connection of graphs

The complexit y of dete rmining the rain b o w v ertex-connection of graphs ∗ Lily Chen, Xueliang Li, Y ongta ng Shi Cen ter for Combinatoric s and LPMC-TJKLC Nank ai Univ ersit y , Tianjin 300071, C h ina Email: lily6061 2@126.co m, lxl@nank ai.edu.cn, sh i@nank ai.edu.cn Abstract A vertex-c olored graph is r ainb ow vertex-c onne cte d if an y t w o ve rtices are con- nected b y a path wh ose inte rnal v ertices ha ve distinct colors, whic h w as int ro duced b y K riv elevic h and Y uster. The r ainb ow v ertex-c onne ction of a connected graph G , denoted by rv c ( G ) , is the smallest n um b er of colors that are needed in order to mak e G rain b o w v ertex-connected. In this pap er, w e study the computational com- plexit y of v ertex-rain b o w connection of graph s and pro ve that computing r v c ( G ) is NP-Hard. Moreo v er, w e sh o w that it is already NP-Complete to decide whether r v c ( G ) = 2. W e also pro v e th at th e follo wing pr oblem is NP-Complete: giv en a v er tex-colored graph G , c hec k wh ether the giv en coloring mak es G rain b o w v ertex- connected. Keyw ords: coloring; rain b o w v ertex-connectio n; compu tational complexit y AMS Sub ject Classification (2010): 05C15, 05C40, 68Q25, 68R10. 1 In tro duction All g raphs considere d in this pap er are simple, finite and undirected . W e follow the notation and terminolog y of Bondy and Murty [1 ]. An edge-colored graph is r ainb ow c onn e c te d if an y tw o v ertices are connected by a path whose edges ha v e distinct colors. This concept of r a in b ow connection in graphs w as in tr o duced b y Chartrand et al. in [4]. The r ainb o w c onn e ction numb e r of a connected ∗ Suppo rted by NSF C and “the F undamental Resear c h F unds for the Central Universities”. 1 graph G , denoted by r c ( G ), is the smallest n um b er of colors that are needed in order to mak e G rain b o w connected. Observ e that diam ( G ) ≤ r c ( G ) ≤ n − 1, where diam ( G ) denotes the diameter of G . It is easy to verify that r c ( G ) = 1 if and only if G is a complete graph, that r c ( G ) = n − 1 if and only if G is a tree. There are some approac hes to study the b ounds of r c ( G ), we refer to [2, 5, 7 ]. In [5], Kriv elevic h a nd Y uster prop osed the concept of rainbow v ertex-connection. A v ertex-colored graph is r ai nb o w vertex-c onne cte d if any t wo v ertices are connected b y a path whose inte rnal v ertices ha ve distinct colors. The r ainb ow vertex-c onne ction of a connected gra ph G , denoted by r v c ( G ), is the smallest n um b er of colors that are needed in o r der to make G rainbow v ertex-connected. An easy observ ation is that if G is of order n then r v c ( G ) ≤ n − 2 and r v c ( G ) = 0 if and o nly if G is a complete graph. Notice tha t r v c ( G ) ≥ diam ( G ) − 1 with equalit y if the diameter is 1 or 2. F or rainbow connection and rain b o w vertex -connection, some examples are giv en to sho w that there is no upp er b ound for one of para meters in t erms of the other in [5]. Kriv elevic h and Y uster [5] prov ed that if G is a g raph with n ve rtices and minim um degree δ , then r v c ( G ) < 11 n/δ . In [6], the authors improv ed this b ound f or giv en order n and minimum degree δ . Besides its t heoretical interest a s b eing a natural com binatoria l concept, rain b o w con- nectivit y can also find applications in net w orking. Suppose w e w ant to route mess ages in a cellular netw o r k suc h that eac h link on the route b et w een tw o vertice s is assigned with a distinct c ha nnel. The minim um n um b er of use d c hannels is exactly the rain b ow connection of the underlying graph. The computational complexit y of rain b ow connection has b een studied. In [2], Caro et al. conjectured that computing r c ( G ) is an NP-Hard problem, as w ell a s that ev en deciding whether a graph has r c ( G ) = 2 is NP-Complete. In [3], Chakrab ort y et al. confirmed this conjecture. Motiv ated b y the pro of of [3], w e consider the computational complexit y of rainbow ve rtex-connection r v c ( G ) of graphs. It is not hard to imag e that this problem is a lso NP-hard, but a rigo rous pro of is necessary . This pap er is to giv e suc h a pro of, whic h follows a similar idea of [3], but b y reducing 3-SA T problem to some other new problems, t ha t computing r v c ( G ) is NP-Hard. Moreo ver, w e sho w that it is already NP-Complete to decide whether r v c ( G ) = 2. W e also pro v e tha t the following problem is NP-Complete: giv en a v ertex-colored graph G , c hec k whether the giv en coloring mak es G rain b o w v ertex-connected. 2 Rain b o w v erte x-connecti on. F or t w o problems A a nd B , we write A  B , if problem A is polynomially reducible to problem B . No w, w e giv e our first theorem. 2 Theorem 1 The fol lowing pr oblem is NP-Complete: given a vertex-c olor e d gr a ph G , che ck whether the given c ol oring m akes G r ai nb o w vertex-c onne cte d. No w w e define Problem 1 and Problem 2 as follows. W e will pro v e Theorem 1 b y reducing Problem 1 to Problem 2, and then 3 - SA T problem t o Problem 1. Problem 1 s − t rain b o w v ertex-connection. Giv en: V ertex-colored graph G with tw o v ertices s, t . Decide: Whether there is a rain b ow vertex -connected path b et w een s and t ? Problem 2 Rain b ow vertex -connection. Giv en: V ertex-colored graph G . Decide: Whether G is rainbow vertex -connected under the coloring? Lemma 1 Pr oblem 1  Pr oblem 2. Pr o of. Giv en a v ertex-colored gra ph G with tw o vertice s s and t . W e w ant to construct a new graph G ′ with a v ertex coloring suc h that G ′ is rain b o w vertex -connected if and only if there is a ra in b ow vertex -connected path from s to t in G . Let V = { v 1 , v 2 , . . . , v n − 1 , v n } b e v ertices of G , where v 1 = s and v n = t . W e construct G ′ as follows . Set V ′ = V ∪ { s ′ , t ′ , a, b } and E ′ = E ∪ { s ′ s, t ′ t } ∪ { av i , bv i : i ∈ [ n ] } . Let c b e the v ertex coloring of G , w e define the v ertex coloring c ′ of G ′ b y c ′ ( v i ) = c ( v i ) for i ∈ { 2 , 3 , . . . , n − 1 } , c ′ ( s ) = c ′ ( a ) = c 1 , c ′ ( t ) = c ′ ( b ) = c 2 , where c 1 , c 2 are the tw o new colors. Supp ose c ′ mak es G ′ rain b o w v ertex-connected. Since each path Q fro m s ′ to t ′ m ust go through s and t , Q can not contain a and b as c ′ ( s ) = c ′ ( a ) = c 1 and c ′ ( t ) = c ′ ( b ) = c 2 . Therefore, an y r a in b ow v ertex-connected path from s ′ to t ′ m ust con tain a rain b o w v ertex- connected path from s to t in G . Th us there is a rainbow vertex -connected path fro m s to t in G under the coloring c . No w assume tha t there is a ra inbow v ertex-connected path from s to t in G under the coloring c . T o pro v e that G ′ is rain b o w v ertex-connected. First, the rain b o w v ertex- connected path from s ′ to v i can b e formed b y g o ing throug h s and b , then to v i for i ∈ { 2 , 3 , . . . , n } . The rainbow v ertex-connected path from s ′ to t ′ can go through s and t and a rainbow v ertex-connected path from s to t in G . The rain b o w v ertex-connected 3 path from t ′ to v i can b e formed b y going through t and a , then to v i for i ∈ { 2 , 3 , . . . , n } . F or the other pair s of vertic es, there is a path b et w een them with length less than 3, th us they are obv ious rainbow vertex -connected. Lemma 2 3 -SA T  Pr oblem 1. Pr o of. Let φ b e an instance of 3- SA T with clauses c 1 , c 2 , . . . , c m and v aria bles x 1 , x 2 , . . . , x n . W e construct a graph G φ with sp ecial vertice s s and t . First, w e intro duce k new v ertices v j 1 , v j 2 , . . . , v j k for eac h x j ∈ c i and ℓ new vertice s v j 1 , v j 2 , . . . , v j ℓ for eac h x j ∈ c i . Without loss o f generality , w e assume that k ≥ 1 and ℓ ≥ 1, otherwise φ can b e simplified. Next, fo r eac h v j a , a ∈ [ k ], w e in tro duce ℓ new v ertices v j a 1 , v j a 2 , . . . , v j aℓ , whic h f orm a path in this order. Similarly , for eac h v j b , b ∈ [ ℓ ], w e in tro duce k new v ertices v j 1 b , v j 2 b , . . . , v j k b , whic h for m a path in that o rder. T herefore, for x j ∈ c i , there are k paths o f length ℓ − 1, and for x j ∈ c i , there are ℓ paths of length k − 1. F or eac h pat h Q in c i ( i ∈ [ m ]), w e join the original v ertex o f Q to the terminal v ertices o f all paths in c i − 1 , where c 0 is the v ertex s . And for eac h path in c m , w e join its t erminal v ertex to t . Th us, a new graph G φ is obtained. No w we define a v ertex coloring of G φ . F or ev ery v ariable x j , w e intro duce k × ℓ distinct colors α j 1 , 1 , α j 1 , 2 , . . . , α j k ,ℓ . W e color v ertices v j a 1 , v j a 2 , . . . , v j aℓ with colors α j a, 1 , α j a, 2 , . . . , α j a,ℓ , resp ectiv ely , and color v j 1 b , v j 2 b , . . . , v j k b with colors α j 1 ,b , α j 2 ,b , . . . , α j k ,b , resp ectiv ely , where a ∈ [ k ] and b ∈ [ ℓ ]. If G φ con ta ins a rain b o w v ertex-connected s − t path Q , then Q must contain one of the newly built paths in eac h c i , i ∈ [ m ], and the path v j a 1 v j a 2 . . . v j aℓ and v j 1 b v j 2 b . . . v j k b can no t b oth app ear in Q . If v j a 1 v j a 2 . . . v j aℓ app ears in Q , set x j = 1 , a nd if v j 1 b v j 2 b . . . v j k b app ears in Q , set x j = 0. Clearly , w e ha ve φ = 1 in this assignmen t. 3 Rain b o w v erte x-connecti on 2 . Before pro ceeding, w e first define three problems. Problem 3 Ra in b ow vertex -connection 2. Giv en: G raph G = ( V , E ). Decide: Whether there is a v ertex coloring of G w ith t w o colors suc h that all pairs ( u, v ) ∈ V ( G ) × V ( G ) a re rainbow verte x-connected? Problem 4 Subset rain b o w vertex -connection 2. 4 Giv en: G raph G = ( V , E ) and a set o f pairs P ⊆ V ( G ) × V ( G ) . Decide: Whether there is a v ertex coloring of G w ith t w o colors suc h that all pairs ( u, v ) ∈ P are ra in b ow vertex -connected? Problem 5 D ifferen t subsets rain b o w v ertex-connection 2. Giv en: Graph G = ( V , E ) and tw o disjoin t subs ets V 1 , V 2 of V w ith a one to one corre- sp onding f : V 1 → V 2 . Decide: Whether there is a v ertex coloring o f G with tw o colors suc h that G is rain b o w v ertex-connected and for each v ∈ V 1 , v and f ( v ) a re assigned differen t colors. In the fo llowing, w e will reduce Problem 4 to Problem 3 and then reduce Problem 5 to Problem 4. F inally , w e will show it is NP-Complete to decide whether r v c ( G ) = 2 b y reducing 3-SA T problem to Problem 3 . Lemma 3 Pr oblem 4  Pr oblem 3. Pr o of. Giv en a graph G = ( V , E ) and a set of pairs P ⊆ V ( G ) × V ( G ) , we construct a graph G ′ = ( V ′ , E ′ ) as follows . F or eac h v ertex v ∈ V , w e introduce a new vertex x v ; for ev ery pa ir ( u, v ) ∈ ( V × V ) \ P , w e in tro duce t w o new v ertices x 1 ( u,v ) and x 2 ( u,v ) ; w e also add tw o new v ertices s, t . Set V ′ = V ∪ { x v : v ∈ V } ∪ { x 1 ( u,v ) , x 2 ( u,v ) : ( u, v ) ∈ ( V × V ) \ P } ∪ { s, t } and E ′ = E ∪ { v x v : v ∈ V } ∪ { u x 1 ( u,v ) , x 1 ( u,v ) x 2 ( u,v ) , x 2 ( u,v ) v : ( u , v ) ∈ ( V × V ) \ P }∪ { s x 1 ( u,v ) , tx 2 ( u,v ) : ( u, v ) ∈ ( V × V ) \ P } ∪ { sx v , tx v : v ∈ V } . Observ e that G is a subgraph of G ′ . In the follow ing, we will pro ve that G ′ is 2-rain b o w v ertex-connected if and only if there is a v ertex coloring o f G with t w o colors suc h that all pairs ( u, v ) ∈ P a r e rain b o w v ertex-connected. No w supp ose there is a v ertex coloring of G ′ with tw o colors whic h mak es G ′ rain b o w v ertex-connected. F or eac h pair ( u , v ) ∈ P , the paths of length no more than 3 that connects u and v hav e to b e in G . Th us, with the coloring all pairs in P are r a in b ow v ertex-connected. On the other hand, let c : V → { 1 , 2 } b e one coloring of G suc h that all pair s ( u, v ) ∈ P are rain b o w vertex -connected. W e extend the coloring as fo llo ws: c ( x v ) = 1 for a ll v ∈ P , c ( x 1 ( u,v ) ) = 1 and c ( x 2 ( u,v ) ) = 2 for a ll ( u, v ) ∈ ( V × V ) \ P , c ( s ) = c ( t ) = 2. W e can see that G ′ is indeed rain b ow v ertex-connected under this coloring. Lemma 4 Pr oblem 5  Pr oblem 4. 5 Pr o of. Giv en a graph G = ( V , E ) a nd tw o disjoin t subsets V 1 , V 2 of V with a one to one corresp onding f . Assume that V 1 = { v 1 , v 2 , . . . , v k } a nd V 2 = { w 1 , w 2 , . . . , w k } satisfying that w i = f ( v i ) for each i ∈ [ k ]. W e construct a new gra ph G ′ = ( V ′ , E ′ ) as fo llows. W e in tro duce six new v ertices x 1 v i w i , x 2 v i w i , x 3 v i w i , x 4 v i w i , x 5 v i w i , x 6 v i w i for eac h pa ir ( v i , w i ), i ∈ [ n ]. W e add a new verte x s . Set V ′ = V ∪ { x j v i w i : i ∈ [ k ] , j ∈ [6] } ∪ { s } , and E ′ = E ∪ { sx 5 v i w i , x 5 v i w i v i , v i x 1 v i w i , x 1 v i w i x 2 v i w i , x 2 v i w i x 3 v i w i , x 3 v i w i x 4 v i w i , x 4 v i w i w i , w i x 6 v i w i , x 6 v i w i s : i ∈ [ k ] } . W e define P b y: P = { ( u, v ) : u, v ∈ V } ∪ { ( x 5 v i w i , x 2 v i w i ) , ( v i , x 3 v i w i ) , ( x 1 v i w i , x 4 v i w i ) , ( x 2 v i w i , w i ) , ( x 3 v i w i , x 6 v i w i ) : i ∈ [ k ] } . Supp ose there is a v ertex coloring of G ′ with t w o colors suc h that a ll pair s ( u, v ) ∈ P are rain b o w ve rtex-connected. Observ e that G is a subgraph of G ′ . F or all ( u, v ) ∈ V × V , they are b elong to P and the paths connect them with length no more t han 3 are b elong to G , thus G is ra in b ow v ertex-connected. W e hav e c ( v i ) 6 = c ( w i ), since { ( x 5 v i w i , x 2 v i w i ) , ( v i , x 3 v i w i ) , ( x 1 v i w i , x 4 v i w i ) , ( x 2 v i w i , w i ) , ( x 3 v i w i , x 6 v i w i ) : i ∈ [ k ] } are rain b o w v ertex- connected in G ′ . On the ot her hand, if there is a 2-v ertex colo r ing c of G suc h that G is r a in b ow vertex - connected and v i , w i are colored differen tly , w e color G ′ with coloring c ′ as follo ws. F or v ∈ V , c ′ ( v ) = c ( v ). If c ( v i ) = 1 , c ( w i ) = 2, then c ′ ( x 1 v i w i ) = c ′ ( x 3 v i w i ) = 2 , c ′ ( x 2 v i w i ) = c ′ ( x 4 v i w i ) = 1 . If c ( v i ) = 2 , c ( w i ) = 1, then c ′ ( x 1 v i w i ) = c ′ ( x 3 v i w i ) = 1 , c ′ ( x 2 v i w i ) = c ′ ( x 4 v i w i ) = 2 . F or all other v ertices, we assign them b y color 1 or 2 a r bitr arily . It is easy to che c k that all ( u, v ) ∈ P a r e rain b o w v ertex-connected. Lemma 5 3 -SA T  Pr oblem 5. Pr o of. Let φ b e an instance of 3- SA T with clauses c 1 , c 2 , . . . , c m and v aria bles x 1 , x 2 , . . . , x n . W e construct a new gra ph G φ and define t w o disjoin t v ertex sets with a one to one cor- resp onding f . Add tw o new v ertices s, t . Set V φ = { c i : i ∈ [ m ] } ∪ { x i , x i : i ∈ [ n ] } ∪ { s, t } and E φ = { c i c j : i, j ∈ [ m ] } ∪ { tx i , t x i : i ∈ [ n ] } ∪ { x i c j : x i ∈ c j } ∪ { x i c j : x i ∈ c j } ∪ { s t } . 6 W e define V 1 = { x 1 , x 2 , . . . , x n } , V 2 = { x 1 , x 2 , . . . , x n } and f : V 1 → V 2 satisfying t hat f ( x i ) = x i . No w we sho w that G φ is 2-rainbow v ertex-connected with different colors b et ween x i and x i if and only if φ is satisfiable. Supp ose there is a v ertex coloring c : V φ → { 0 , 1 } suc h that G φ is rain b ow v ertex- connected a nd x i , x i are colored differently . W e first supp ose c ( t ) = 0, set the v alue of x i as the corresponding color of x i . F or eac h i , consider the rain b o w v ertex-connected path Q b et wee n v ertices s and c i , there m ust exist some j suc h that w e can write Q = stx j c i or Q = st x j c i . Without loss of generality , supp ose Q = stx j c i . Since c ( t ) = 0, w e hav e c ( x j ) = 1. Th us, the v a lue of x j is 1, whic h concludes that c i = 1 as x j ∈ c i b y the construction of G φ . F or the other case c ( t ) = 1 , w e set x i = 1 if c ( x i ) = 0 a nd x i = 0 otherwise. By some similar discuss ions, w e also hav e φ = 1. On the other hand, for a given trut h assignmen t o f φ , we colo r G φ as follo ws: c ( t ) = 0 and c ( c i ) = 1 fo r i ∈ [ m ]; if x i = 1, then c ( x i ) = 1 and c ( x i ) = 0; otherwise, c ( x i ) = 0 and c ( x i ) = 1. W e can easily chec k that G φ is rain b o w v ertex-connected. F rom the ab o v e three lemmas, w e conclude our second theorem. Theorem 2 Given a gr aph G , de ciding whether r v c ( G ) = 2 is NP-Complete. Thus, c omputing r v c ( G ) is NP-Har d. References [1] J.A. Bo ndy and U.S.R. Murty , Graph Theory , GTM 2 44, Springer, 2008. [2] Y. Caro, A. Lev, Y. Ro ditt y , Z. T uza and R. Y uster, On r a in b ow connection, Ele ctr o n J. Combin. 15 (2008), R57. [3] S. Chakrab o r ty , E. Fisc her, A. Matsliah and R. Y uster, Hardness and algorithms for rain b o w connectivit y , J. Comb. Optim. , in press. [4] G. Chartra nd, G.L. Johns, K.A. McKeon a nd P . Zhang, Rainbow connection in graphs, Math. Bohemic a 133 (2008), 85–98 . [5] M. Kriv elevic h and R. Y uster, The rainbow connection o f a g r a ph is (a t most) recip- ro cal to its minim um degree, J. Gr a ph The ory 63 (2010), 185 –191. [6] X. Li a nd Y. Shi, On the r a in b ow vertex -connection, arXiv:1012.3 504. [7] I. Sc hiermey er, Rainbow connection in graphs with minim um degree three, IW OCA 2009, LNCS 5874 (2009 ) , 43 2–437. 7