A Lower Bound of the cd-Chromatic Number and Its Complexity

Abstract

The cd-coloring is motivated by the super-peer architecture in peer-to-peer resource sharing technology. A vertex set partition of a graph G into k independent sets $V_1, V_2, \ldots , V_k$ is called a k-color domination partition (k-cd-coloring) of G if there exists a vertex $u_i\in V(G)$ such that $u_i$ dominates $V_i$ in G for $1 \le i \le k$ and the smallest integer k for which G admits a k-cd-coloring is called the cd-chromatic number of G, denoted by $\chi _{cd}(G)$. A subclique is a set S of vertices of a graph G such that for any $x,y\in S$, $d(x,y)\ne 2$ in G and the cardinality of a maximum subclique in G is denoted by $\omega _{s}(G)$. Clearly, $\omega _{s}(G) \le \chi _{cd}(G)$ for a graph G.

In this paper, we explore the complexity status of Subclique: for a given graph G and a positive integer k, Subclique is to decide whether G has a subclique of size at least k. We prove that Subclique is NP-complete for (i) bipartite graphs, (ii) chordal graphs, and (iii) the class of H-free graphs when H is a fixed graph on 5-vertices. In addition, we prove that Subclique for the class of H-free graphs is polynomial time solvable only if H is an induced subgraph of $P_4$; otherwise the problem is NP-complete. Moreover, Subclique is polynomial time solvable for trees, split graphs, and co-bipartite graphs.