Relating the super domination and 2-domination numbers in cactus graphs

A set D subset of V(G) is a super dominating set of a graphG if for every vertex u is an element of V(G)\, there exists a vertex v is an element of D such that N(v)\D = {u}. The super domination number ofG, denoted by gamma(sp)(G), is the minimum cardinality among all super dominating sets of G. In this article, we show that if G is a cactus graph with k(G) cycles, then gamma(sp)(G) <=gamma(2)(G) + k(G), where gamma(2)(G) is the 2-domination number of G. In addition, and as a consequence of the previous relationship, we show that if T is a tree of order at least three, then gamma(sp)(T) <= alpha(T) + s( T) - 1 and characterize the trees attaining this bound, where alpha(T) and s(T) are the independence number and the number of support vertices of T, respectively.