Decomposition of graphs: some polynomial cases

Cristina Bazgan (1) , Zsolt Tuza (2) , Daniel Vanderpooten (1)

1 LAMSADE - Laboratoire d'analyse et modélisation de systèmes pour l'aide à la décision
2 MTA SZTAKI - Computer and Automation Research Institute [Budapest]

We study the problem of decomposing the vertex set V of a graph into two parts (V1,V2) which induce subgraphs where each vertex v in V1 has degree at least a(v) and each vertex v in V2 has degree at least b(v). We investigate several polynomial cases of this NP-complete problem. We give a polynomial-time algorithm forgraphs with bounded treewidth which decides if a graph admits a decomposition and gives such a decomposition if it exists. We also give polynomial-time algorithms that always find a decomposition for the following two cases : triangle-free graphs such that d(v)>= a(v)+b(v) for every vertex v in V and graphs with girth at least 5 such that d(v)>= a(v)+b(v)-1 for every vertex v in V.

Mots clés: Graph; decomposition; degree constraints; treewidth; girth; complexity; polynomial algorithm

Domaines: Complexité [cs.CC]; Mathématique discrète [cs.DM]

Cristina Bazgan, Zsolt Tuza, Daniel Vanderpooten. Decomposition of graphs: some polynomial cases. 2005. ⟨hal-00004259⟩