Dominating number of distance two of corona products of graphs

Abstract

Dominating set $S$ in graph $G=(V,E)$ is a subset of $V(G)$ such that every vertex of $G$ which is not element of $S$ are connected and have distance one to $S$. Minimum cardinality among dominating sets in a graph $G$ is called dominating number of graph $G$ and denoted by $\gamma(G)$. While dominating set ofdistance two which denoted by $S_2$ is a subset of $V(G)$ such that every vertex of $G$ which is not element of $S$ are connected and have maximum distance two to $S_2$. Dominating number of distance two $\gamma_2(G)$ is minimum cardinality of dominating set of distance two $S_2$. The corona $G \odot H$ of two graphs $G$ and $H$ where $G$ has $p$ vertices and $q$ edges is defined as the graph G obtained by taking one copy of $G$ and $p$ copies of $H$, and then joining by an edge the $i-th$ vertex of $G$ to every vertex in the $i-th$ copy of $H$. In this paper, we determine the dominating number of distance two of paths and cycles. We also determine the dominating number of distance two of corona product of path and any graphs as well as cycle and any graphs.