Enrico Enriquez, Jonecis Dayap
Let G = (V (G), E(G)) be a simple connected graph. A dominating set S in G is called a secure dominating set in G if for every u∈V (G) \S, there exists v ∈ S n NG(u) such that (S \ {v}) u {u} is a dominating set. The minimum cardinality of secure dominating set is called the securedomination number of G and is denoted by ys(G). A secure dominatingset of cardinality ys(G) is called ys-set of G. Let D be a minimum secure dominating set in G. The secure dominating set S ⊆ V (G) \ D is called an inverse secure dominating set with respect to D. The inversesecure domination number of G denoted by ys-1(G)is the minimumcardinality of an inverse secure dominating set in G. An inverse secure dominating set of cardinality ys-1(G) is called ys-1-set. A disjoint secure dominating set in G is the set C = D ∪ S ⊆ V (G). The disjoint securedomination number of G denoted by yyr(G)is the minimum cardinalityof a disjoint secure dominating set in G. A disjoint secure dominating set of cardinality yys(G) is called yys-set. In this paper, we show that every integers k and n with k∈ {2, 4, 5, …n - 1, n} is realizable as disjoint secure domination number, and order of G respectively. Further, we give the characterization of the disjoint secure dominating set in the join of two graphs. © 2016, University of San Jose-Recoletos. All rights reserved.
University of San Carlos, Philippines; University of San Jose-Recoletos, Philippines
