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We proveτ ≤ 2 7/9v for 3-partite hypergraphs. (This is an improvement of the trivial boundτ ≤ 3v.)

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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 graphsGG that have a cut-vertex or do not contain any cutsetS dominated by somes∈S.

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