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

Abstract

A set D⊆V(G) is a super dominating set of a graph G if for every vertex u∈V(G)\D , there exists a vertex v∈D such that N(v)\D={u} . The super domination number of G , denoted by γsp(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 γsp(G)≤γ2(G)+k(G) , where γ2(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 γsp(T)≤α(T)+s(T)−1 and characterize the trees attaining this bound, where α(T) and s(T) are the independence number and the number of support vertices of T , respectively.