FORCING SUPER DOMINATION NUMBER OF A GRAPH

Let G = (V(G), E(G)) be a simple graph and let X subset of V(G). A vertex v is an element of V(G)\X is an external private neighbor of u is an element of X subset of V(G) with respect to X if N-G (v) boolean AND X = {u}. A set S subset of V(G) is a dominating set if every vertex u is an element of V(G)\S is adjacent to at least one vertex of S. A dominating set S of G is a super dominating set if every vertex v is an element of V(G)\S has an external private neighbor with respect to V(G)\S. The super domination number of G, denoted by gamma(sp) (G), is the minimum cardinality of a super dominating set in G. A super dominating set S of G with vertical bar S vertical bar = gamma(sp) (G) is called a gamma(sp)-set of G. A subset D of a gamma(sp)-set is a forcing subset for S if S is the only gamma(sp)- set of G containing D. The forcing super domination number of S is given by f(G)(S, gamma(sp)) = min{vertical bar D vertical bar: D is a forcing subset for S}. The forcing super domination number of G is given by f(G, gamma(sp) ) = min{f(G)(S, gamma(sp)) : S is a gamma(sp)-set of G}. In this paper, we determine the super domination number of some common graphs and the join of some graphs.