Maximum max-k-clique subgraphs in cactus subtree graphs

Abstract

In this paper, we analyze the intersection graphs of subtrees in a cactus, called cactus subtree graphs. A max-k-clique subgraph of a graph is an induced subgraph whose maximum clique has at most [Formula: see text] vertices. We describe a polynomial time algorithm to find a maximum weight max-[Formula: see text]-clique subgraph in a cactus subtree graph when [Formula: see text] is fixed. In perfect graphs this problem is equivalent to the maximum weight [Formula: see text]-colorable subgraph problem. In many applications like scheduling production lines and communication networks on imperfect graphs, a maximum max-[Formula: see text]-clique subgraph solution is better than a maximum [Formula: see text]-colorable subgraph solution. In addition, we describe cactus subtree graphs polynomial time algorithms for recognition, maximum independent sets, maximum cliques, maximum induced bipartite graphs, maximum induced forests and minimum feedback vertex sets.