Full domination in graphs

For each vertex v in a graph G, let there be associated a subgraph ${\displaystyle H_v}$ of G. The vertex v is said to dominate ${\displaystyle H_v}$ as well as dominate each vertex and edge of ${\displaystyle H_v}$. 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 ${\displaystyle {\gamma }_{FH}\left(G\right)}$. A full dominating set of G of cardinality ${\displaystyle {\gamma }_{FH}\left(G\right)}$ is called a ${\displaystyle {\gamma }_{FH}}$-set of G. We study three types of full domination in graphs: full star domination, where ${\displaystyle H_v}$ is the maximum star centered at v, full closed domination, where ${\displaystyle H_v}$ is the subgraph induced by the closed neighborhood of v, and full open domination, where ${\displaystyle H_v}$ is the subgraph induced by the open neighborhood of v.