Outer-Clique Domination in Graphs

Enrico Limbo Enriquez, Jovita N. Ravina, Valerie Verallo Fernandez


Let  be a simple graph. A set of vertices of a graph is an outer-clique dominating set if every vertex not in  is adjacent to some vertex inand the subgraph induced by is clique. In this paper, we will show that given positive integers  andsuch thatthere exists a connected nontrivial graph  with and  Further, we give characterization the outer-clique dominating sets resulting from the join of two graphs and give some important results.  In this paper, we show that for each set of integers and  with the integers  and  are realizable as domination number, outer-clique domination number, and order of , respectively. Further, we give the characterization of the outer-clique dominating set with outer-clique domination numbers of 1 and 2. Finally, we characterize outer-clique dominating sets of the join of two graphs.

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