One-factor Resolvability of Grid derived Networks

Abstract

Given a graph $G=(V,E)$, a set $W\subseteq V$ is said to be a resolving set if for each pair of distinct vertices $u,v\in V$, there is a vertex $x$ in $W$ such that $d(u,x)\ne d(v,x)$. The resolving number of $G$ is the minimum cardinality of all resolving sets. In this paper, a condition is imposed on resolving sets and a conditional resolving parameter is studied for grid-based networks.