On the complexity of coloring ‐graphs
Matheus S. D. Alves, Julliano R. Nascimento, Uéverton S. Souza · International Transactions in Operational Research · 2021
Abstract An ‐graph is a graph that can be partitioned into r independent sets and ℓ cliques. In the Graph Coloring problem, we are given as input a graph G, and the objective is to determine the minimum integer k such that G admits a proper vertex k‐coloring. In this work, we describe a Poly versus NP‐hard dichotomy of this problem regarding the parameters r and ℓ of ‐graphs, which determine the boundaries of the ‐hardness of Graph Coloring for such classes. We also analyze the complexity of the problem on ( )‐graphs under the parameterized complexity perspective. We show that given a (2, 1)‐partition of a graph G, finding an optimal coloring of G is ‐complete even when K1 is a maximal clique of size 3; XP but W[1]‐hard when parameterized by ; fixed‐parameter tractable (FPT) and admits a polynomial kernel when parameterized by . Besides, concerning the case where K1 is a maximal clique of size 3, a P versus NPh dichotomy regarding the neighborhood of K1 is provided; furthermore, an FPT algorithm parameterized by the number of vertices having no neighbor in K1 is presented.