Inverse Domination Number and Inverse Total Domination of Sierpinski Star Graph

Abstract

Given a graph G=(V(G),E(G)) consisting of the set of vertices V(G) and the set of edges E(G). For example, D(G) is a domination set of graph G with minimum cardinality, if V(G)-D(G) contains a domination set D^(-1) (G), then D^(-1) (G) is called the inverse domination set of graph G. The minimum cardinality of the inverse domination set of the graph G is called the inverse domination number, denoted by γ^(-1) (G). If D_t (G) is the total domination set of the graph G with minimal cardinality, and V(G)-D_t (G) contains the total domination set D_t^(-1) (G), then D_t^(-1) (G) is called the inverse total domination set of the graph G. The minimum cardinality of the inverse total domination set of the graph G is called the inverse total domination number, denoted by γ_t^(-1) (G). This paper discusses the inverse domination and the inverse total domination on the Sierpinski Star graph SS_n, obtained the inverse domination number γ^(-1) (SS_n )=0 for n<3 and γ^(-1) (SS_n )=4∙3^(n-3) for n≥3 and the inverse total domination number γ_t^(-1) (SS_n )=0 for n≥1.