Philippine E-Journals

Disjoint Secure Dominationin the Join of Graphs

Enrico Enriquez | Jonecis Dayap

 

Abstract:

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 ∩ NG(u) such that (S {v}) ∪ {u} is a dominating set. The minimum cardinality of secure dominating set is called the secure domination number of G and is denoted by γs(G). A secure dominating set of cardinality γs(G) is called γs-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 inverse secure domination number of G denoted by γs−1(G)is the minimum cardinality of an inverse secure dominating set in G. An inverse secure dominating set of cardinality γs−1(G) is called γs−1-set. A disjoint secure dominating set in G is the set C = D∪S⊆V (G). The disjoint secure domination number of G denoted by γγr(G)is the minimum cardinality of a disjoint secure dominating set in G. A disjoint secure dominating set of cardinality γγs(G) is called γγs-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.

Mathematics Subject Classification: 05C69