The distance-$k$ dimension of graphs

Jesse Geneson, Eunjeong Yi

Published 2021-06-15Version 1

The metric dimension, $\dim(G)$, of a graph $G$ is a graph parameter motivated by robot navigation that has been studied extensively. Let $G$ be a graph with vertex set $V(G)$, and let $d(x,y)$ denote the length of a shortest $x-y$ path in $G$. For a positive integer $k$ and for distinct $x,y \in V(G)$, let $d_k(x,y)=\min\{d(x,y), k+1\}$ and let $R_k\{x,y\}=\{z\in V(G): d_k(x,z) \neq d_k(y,z)\}$. A subset $S\subseteq V(G)$ is a distance-$k$ resolving set of $G$ if $|S \cap R_k\{x,y\}| \ge 1$ for any pair of distinct $x,y \in V(G)$, and the distance-$k$ dimension, $\dim_k(G)$, of $G$ is the minimum cardinality over all distance-$k$ resolving sets of $G$. Note that, for the diameter $d$ of $G$, $\dim_{k}(G)=\dim(G)$ if $k \ge d-1$, and $\dim_1(G)$ is the adjacency dimension of $G$ introduced by Jannesari and Omoomi in 2012. In this paper, we initiate the study of distance-$k$ dimension of graphs. We obtain some general bounds for distance-$k$ dimension. We characterize connected graphs $G$ of order $n$ with $\dim_k(G) \in \left\{1, n-2, n-1 \right\}$ for all $k \ge 1$. We determine $\dim_k(G)$ when $G$ is a cycle or a path. We also examine the effect of vertex or edge deletion on the distance-$k$ dimension of graphs.