Determining sets, resolving sets, and the exchange property

TL;DR

This paper investigates the exchange property of determining and resolving sets in graphs, showing it holds in trees and certain outerplanar graphs but not universally, and explores specific graph families like n-wheels.

Contribution

It characterizes when determining and resolving sets have the exchange property across various graph classes, including trees, outerplanar graphs, and n-wheels.

Findings

Exchange property holds in trees for both sets.

Determining sets have exchange property in n-wheels with n≥8.

Necessary and sufficient conditions identified for outerplanar graphs.

Abstract

A subset U of vertices of a graph G is called a determining set if every automorphism of G is uniquely determined by its action on the vertices of U. A subset W is called a resolving set if every vertex in G is uniquely determined by its distances to the vertices of W. Determining (resolving) sets are said to have the exchange property in G if whenever S and R are minimal determining (resolving) sets for G and r\in R, then there exists s\in S so that S-\{s\}\cup \{r\} is a minimal determining (resolving) set. This work examines graph families in which these sets do, or do not, have the exchange property. This paper shows that neither determining sets nor resolving sets have the exchange property in all graphs, but that both have the exchange property in trees. It also gives an infinite graph family (n-wheels where n\geq 8) in which determining sets have the exchange property but resolving sets do not. Further, this paper provides necessary and sufficient conditions for determining sets to have the exchange property in an outerplanar graph.