On H-dominating matchings and some number partitions

A dominating set of a graph G = (V,E) is a subset D of V such that every vertex of V − D is adjacent to a vertex in D. In this paper we introduce a generalization of domination as follows. For graphs G and H, an H-matching M of G is a subgraph of G such that all components of M are isomorphic to H. An H-dominating matching of G is a Hmatching D of G such that for each x ∈ V (G) there exists y ∈ V (D) such that xy ∈ E(G). We consider Pk-dominating matchings for k ≥ 1. We generalize recent results by Zwierzchowski [Graph Theory Notes of New York, XLVI, NY Acad. Sc. 2004, 13–19] on the number of dominating sets for the graphs Pn and Cn.