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

Abstract

Abstract A set D ⊆ V ( G ) D\subseteq V\left(G) is a super dominating set of a graph G G if for every vertex u ∈ V ( G ) \ D u\in V\left(G)\setminus D , there exists a vertex v ∈ D v\in D such that N ( v ) \ D = { u } N\left(v)\setminus D=\left\{u\right\} . The super domination number of G G , denoted by γ s p ( G ) {\gamma }_{sp}\left(G) , is the minimum cardinality among all super dominating sets of G G . In this article, we show that if G G is a cactus graph with k ( G ) k\left(G) cycles, then γ s p ( G ) ≤ γ 2 ( G ) + k ( G ) {\gamma }_{sp}\left(G)\le {\gamma }_{2}\left(G)+k\left(G) , where γ 2 ( G ) {\gamma }_{2}\left(G) is the 2-domination number of G G . In addition, and as a consequence of the previous relationship, we show that if T T is a tree of order at least three, then γ s p ( T ) ≤ α ( T ) + s ( T ) − 1 {\gamma }_{sp}\left(T)\le \alpha \left(T)+s\left(T)-1 and characterize the trees attaining this bound, where α ( T ) \alpha \left(T) and s ( T ) s\left(T) are the independence number and the number of support vertices of T T , respectively.