Distance $k$-domination and $k$-resolving domination of the corona product of graphs

Abstract

For two simple graphs $G$ and $H$, the corona product of $G$ and $H$ is the graph obtained by adding a copy of $H$ for every vertex of $G$ and joining each vertex of $G$ to its corresponding copy of $H$. For $k \geq 1$, a set of vertices $D$ in a graph $G$ is a distance $k$-dominating set if any vertex in $G$ is at a distance less or equal to $k$ from some vertex in $D$. The minimum cardinality overall distance $k$-dominating sets of $G$ is the distance $k$-domination number, denoted by $\gamma_k(G)$. The metric dimension of a graph is the smallest number of vertices required to distinguish all other vertices based on distances uniquely. The distance $k$-resolving domination in graphs combines distance $k$-domination and the metric dimension of graphs. In this paper, we investigate for all $k\geq 1$, the distance $k$-domination and the distance $k$-resolving domination in the corona product of graphs. First, we show that for $k\geq 2$ the distance $k$-domination number of $G\odot H$ is equal to $\gamma_{k-1}(G)$ for any two graphs $G$ and $H$. Then, we give the exact value of $\gamma_{k}(G\odot H)$ when $G$ is a complete graph, complete $m$-partite graph, path and cycle. We also provide general bounds for $\gamma_{k}(G\odot H)$. Then, we examine the distance $k$-resolving domination number for $G\odot H$. For $k=1$, we give bounds for $\gamma^r(G\odot H)$ the resolving domination number of $G\odot H$ and characterize the graphs achieving those bounds. Later, for $k\geq 2$, we establish bounds for $\gamma^r_k(G\odot H)$ the distance $k$-resolving domination number of $G\odot H$ and characterize the graphs achieving these bounds.