Upper Secure Domination Number of a Graph

A subset S of V   is a dominating set of a graph G=(V,E)G=(V,E) if every vertex in V\SV\S is adjacent to a vertex in S. A dominating set S   is called a secure dominating set if for each v∈V\Sv∈V\S there exists u∈Su∈S such that v is adjacent to u   and S1=(S−{u})∪{v}S1=(S−{u})∪{v} is a dominating set. The maximum cardinality of a minimal secure dominating set of G is called the upper secure domination number of G   and is denoted by Γs(G)Γs(G). In this paper we initiate a study of this parameter and present several basic results.