The k-Metric Dimension of Corona Product Graphs

Abstract

Given a connected simple graph $G=(V,E)$ and a positive integer k, a set $S\subseteq V$ is said to be a k-metric generator for G if and only if for any pair of different vertices $u,v\in V$, there exist at least k vertices $w_1,w_2,\ldots ,w_k\in S$ such that $d_G(u,w_i)\ne d_G(v,w_i)$, for every $i\in \{1,\ldots ,k\}$, where $d_G(x,y)$ is the length of a shortest path between x and y. A k-metric generator of minimum cardinality in G is called a k-metric basis and its cardinality, the k-metric dimension of G. In this article, we study the k-metric dimension of corona product graphs $G\odot \mathcal {H}$, where G is a graph of order n and $\mathcal {H}$ is a family of n non-trivial graphs. Specifically, we give some necessary and sufficient conditions for the existence of a k-metric basis in a connected corona graph. Moreover, we obtain tight bounds and closed formulae for the k-metric dimension of connected corona graphs.