Exact square coloring of certain classes of graphs: Complexity and algorithms

Abstract

A vertex coloring of a graph G=(V,E) is called an exact square coloring of G if any two vertices at distance exactly 2 receive different colors. The minimum number of colors required by an exact square coloring is called the exact square chromatic number of G and is denoted by χ[#2](G). Given a graph G and a positive integer k, the Exact Square Coloring Problem is to decide whether G admits an exact square coloring using k colors. It is known that Exact Square Coloring Problem is NP-complete for chordal graphs. In this paper, we strengthen this result by proving that this problem remains NP-complete for undirected path graphs, which is a proper subclass of chordal graphs. However, we give linear time algorithms for computing the exact square chromatic number of proper interval graphs and threshold graphs, which are proper subclasses of chordal graphs. Moreover, for a proper interval graph G, we show that χ[#2](G)≤3. We also propose a polynomial time algorithm to produce an exact square coloring of a block graph G using at most χ[#2](G)+1 colors. Next, we study a lower bound of χ[#2](G). A subset S of vertices of a graph G=(V,E) is called an exact square clique of G if the distance between any two vertices in S is exactly 2. The cardinality of the maximum exact square clique of G is called the exact square clique number of G and is denoted by ω[#2](G). Clearly, ω[#2](G)≤χ[#2](G). Given a graph G and a positive integer k, the problem of deciding whether ω[#2](G) is at least k, is known to be NP-complete for bipartite graphs and chordal graphs. In this paper, we strengthen these results by proving that this problem remains NP-complete for undirected path graphs, perfect elimination bipartite graphs, star-convex bipartite graphs and comb-convex bipartite graphs. We also compute the exact value of ω[#2](G) for proper interval graphs, threshold graphs, block graphs and convex bipartite graphs.