Coloring Bull-Free Perfect Graphs

Abstract

A graph $G$ is perfect if for every induced subgraph $H$ of $G$, the chromatic number of $H$ equals the size of the largest complete subgraph of $H$. A bull is a graph on five vertices consisting of a triangle and two vertex-disjoint pendant edges. A graph is said to be bull-free if none of its induced subgraphs is a bull. In [SIAM J. Discrete Math., 18 (2004), pp. 226--240], de Figueiredo and Maffray gave polynomial time combinatorial algorithms that solve the following four optimization problems for weighted bull-free perfect graphs with integer weights: the maximum weighted clique problem; the maximum weighted stable set problem; the minimum weighted coloring problem; and the minimum weighted clique covering problem. In this paper, we give faster combinatorial algorithms that solve the same four problems. The running time of our algorithms for finding a maximum weighted clique and a maximum weighted stable set in a weighted bull-free perfect graph with integer weights is $O(n^6)$, and the running time of our algorithms for finding a minimum weighted coloring and a minimum weighted clique covering in such a graph is $O(n^8)$, where $n$ is the number of vertices of the input graph.