Geodesic
Full domination in graphs
Discussiones Mathematicae. Graph Theory, Tome 21 (2001) no. 1, pp. 43-62

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For each vertex v in a graph G, let there be associated a subgraph H_v of G. The vertex v is said to dominate H_v as well as dominate each vertex and edge of 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 γ_FH(G). A full dominating set of G of cardinality γ_FH(G) is called a γ_FH-set of G. We study three types of full domination in graphs: full star domination, where H_v is the maximum star centered at v, full closed domination, where H_v is the subgraph induced by the closed neighborhood of v, and full open domination, where H_v is the subgraph induced by the open neighborhood of v.
Keywords: full domination, full star domination, full closed domination, full open domination 
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Brigham, Robert; Chartrand, Gary; Dutton, Ronald; Zhang, Ping. Full domination in graphs. Discussiones Mathematicae. Graph Theory, Tome 21 (2001) no. 1, pp. 43-62. http://geodesic.mathdoc.fr/item/DMGT_2001_21_1_a3/