Secure domination in rooted product graphs

Abstract

A secure dominating set of a graph G is a dominating set S satisfying that for every vertex $v\in V(G){\setminus } S$ there exists a neighbour $u\in S$ of v such that $(S\cup \{v\}){\setminus } \{u\}$ is a dominating set as well. The secure domination number, denoted by $\gamma _s(G)$, is the minimum cardinality among all secure dominating sets of G. This concept was introduced in 2005 by Cockayne et al. and studied further in a number of works. The problem of computing the secure domination number is NP-Hard. This suggests finding the secure domination number for special classes of graphs or obtaining tight bounds on this invariant. The aim of this work is to obtain closed formulas for the secure domination number of rooted product graphs. We show that for any graph G of order n(G) and any graph H with root v, the secure domination number of the rooted product graph $G\circ _vH$ satisfies one of the following three formulas, $\gamma _{s}(G\circ _vH)=n(G)(\gamma _{s}(H)-1)+\gamma (G)$, $\gamma _{s}(G\circ _vH)= n(G)(\gamma _{s}(H)-1)+\gamma _{s}(G)$ or $\gamma _{s}(G\circ _vH)= n(G)\gamma _{s}(H)$, where $\gamma (G)$ denotes the domination number of G. We also characterize the graphs that satisfy each of these expressions. As a particular case of the study, we derive the corresponding results for corona graphs.