AbstractA K-Cluster in a graph is an induced sub graph on k-vertices, which maximizes the number of edges. Both the K-Cluster problem and the K-dominating set problem are NP-complete for graphs in general. In this paper we investigate the complexity status of these problems on various subclasses of perfect graphs. In particular, we examine comparability graphs, chordal graphs, bipartite graphs, split graphs, co graphs and K-trees. For example, it is shown that the K-Cluster problem is NP-complete for both bipartite and chordal graphs and the independent K-dominating set problem is NP-complete for bipartite graphs. Furthermore, where the Kcluster problem is polynomial we study the weighted and connected versions as well. Similarly we also look at the minimum K-dominating set problem on families, which have polynomial K-dominating set algorithms.
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