Minimum Weight Resolving Sets of Grid Graphs

TL;DR

This paper investigates the minimum weight resolving sets in grid graphs, providing characterizations for small solutions and bounds on maximum solution size, advancing understanding of resolving sets in structured graphs.

Contribution

It offers a complete characterization of small resolving sets and bounds on maximum size for grid graphs, a novel analysis in graph resolving set theory.

Findings

Characterization of resolving sets of size 2 or 3

Maximum size of resolving sets is 2n-2

Bounds on minimal resolving sets from 4 to 2n-2

Abstract

For a simple graph $G=(V,E)$ and for a pair of vertices $u,v \in V$, we say that a vertex $w \in V$ resolves $u$ and $v$ if the shortest path from $w$ to $u$ is of a different length than the shortest path from $w$ to $v$. A set of vertices ${R \subseteq V}$ is a resolving set if for every pair of vertices $u$ and $v$ in $G$, there exists a vertex $w \in R$ that resolves $u$ and $v$. The minimum weight resolving set problem is to find a resolving set $M$ for a weighted graph $G$ such that$\sum_{v \in M} w(v)$ is minimum, where $w(v)$ is the weight of vertex $v$. In this paper, we explore the possible solutions of this problem for grid graphs $P_n \square P_m$ where $3\leq n \leq m$. We give a complete characterisation of solutions whose cardinalities are 2 or 3, and show that the maximum cardinality of a solution is $2n-2$. We also provide a characterisation of a class of minimals whose cardinalities range from $4$ to $2n-2$.