Dominating cliques in P5-free graphs

All induced connected subgraphs of a graphG contain a dominating set of pair-wise adjacent vertices if and only if there is no induced subgraph isomorphic to a path or a cycle of five vertices inG. Moreover, the problem of finding any given type of connected dominating sets in all members of a classG of graphs can be reduced to the graphsG∈G that have a cut-vertex or do not contain any cutsetS dominated by somes∈S. This research was supported in part by the “AKA” Research Fund of the Hungarian Academy of Sciences.