The dominated coloring problem and its application

Abstract

Given a graph G = (V,E), a set Z of V is dominated by a vertex v ε V if v is adjacent to all vertices of Z. A proper coloring in G is an assignment of colors to each vertex of V such that any two adjacent vertices have different colors. A dominated coloring in G is a proper coloring such that each color class is dominated by a vertex in V. The dominated coloring problem is to find a dominated coloring D in G such that the number of color classes in D is minimized. In this paper, we provide the first possible application of the dominated coloring problem that is the development of interpersonal relationships in the social network. Since the dominated coloring problem has been listed to be NP-hard, we prove that the dominated coloring problem has a lower bound of |V|(1-ε)+1/1.5 on the best possible approximation ratio in general graphs, for any ε > 0, and is NP-complete even when the graphs are bipartite graphs. Then we design the first non-trivial brute force exact algorithm to solve the dominated coloring problem in general graphs. Since the dominated coloring problem has a lower bound of |V| 1-ε+1/1.5 on the best possible approximation ratio in general graphs, we present an H(d)-approximation algorithm that is faster but still exponential-time, where d is the maximum degree of G and H(d) is the d-th harmonic number. The H(d)-approximation algorithm can run in polynomial time in bipartite graphs. © 2014 Springer International Publishing.