Full domination in graphs

For each vertex v in a graph G, let there be associated a subgraph Hv of G. The vertex v is said to dominate Hv as well as dominate each vertex and edge of Hv. A set S of vertices of G is called a full dominating set if every vertex of G is dominated by some vertex of S, as is every edge of G. The minimum cardinality of a full dominating set of G is its full domination number γFH(G). A full dominating set of G of cardinality γFH(G) is called a γFH -set of G. We study three types of full domination in graphs: full star domination, where Hv is the maximum star centered at v, full closed domination, where Hv is the subgraph induced by the closed neighborhood of v, and full open domination, where Hv is the subgraph induced by the open neighborhood of v.