Matroids with bases as minimal resolving sets of graphs

TL;DR

This paper introduces a matroid structure based on minimal resolving sets in graphs, enabling efficient algorithms for finding minimum resolving sets in weighted graphs, with specific characterizations for trees.

Contribution

It establishes that the independence system of minimal resolving sets forms a matroid for certain graph families, including trees, and explores properties of the dual matroid.

Findings

Matroid structure exists for minimal resolving sets in trees and some graph families.

Greedy algorithm can find minimum-cost resolving sets using the matroid.

Dual matroid of the constructed matroid is loop-free.

Abstract

We define an independence system associated with simple graphs. We prove that the independence system is a matroid for certain families of graphs, including trees, with bases as minimal resolving sets. Consequently, the greedy algorithm on the matroid can be used to find the minimum-cost resolving set of weighted graphs, wherein the independent system is a matroid. We also characterize hyperplanes of the matroid for trees and prove that its dual matroid is loop-free.