On the complexity of cd-coloring of graphs

Abstract

A partition of the vertex set of a graph G into k independent sets V1,V2,…,Vk is called a k-class domination coloring (k-cd-coloring) of G if for every Vi, 1≤i≤k, there exists a vertex xi such that Vi⊆N[xi]. For a given graph G and a positive integer k, the cd-colorability problem, CD-Colorability, is to decide whether G admits a k-cd-coloring. This problem is NP-complete, even for bipartite graphs. In this paper, we study the complexity of this problem on several natural graph classes.We prove that CD-Colorability is NP-complete for chordal graphs. We also obtain the following dichotomy results: (i) CD-Colorability is NP-complete for graphs with α(G)≥3 and is polynomial time solvable for graphs with α(G)≤2; (ii) CD-Colorability of K1,p-free graphs is NP-complete for p≥4 and is polynomial time solvable for p≤3; (iii) CD-Colorability for the class of H-free graphs is polynomial time solvable if H is an induced subgraph of P4 or P3∪K1 or K1,3 and NP-complete for any other H. We present polynomial time algorithms for the classes of claw-free graphs, P4-free graphs, and double-split graphs. We also characterize the class of all 3-cd-colorable graphs and thereby improve the time complexity of an algorithm of Merouane et al. (2015) for recognizing 3-cd-colorable graphs from O(n6) to O(n5), where n=|V(G)|.