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Matroids with bases as minimal resolving sets of graphs

Usman Ali, Iffat Fida Hussain

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.

Matroids with bases as minimal resolving sets of graphs

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.

Paper Structure

This paper contains 9 sections, 9 theorems, 3 equations, 3 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Metric Dimension
  4. Basics of Matroids
  5. Our Results
  6. Trees
  7. Wheels
  8. Cycles and Complete Graphs
  9. Open Questions

Key Result

Theorem 6

(Minimal resolving set criterion). A set $S$ of vertices is considered a minimal resolving set for a tree if and only if, for each exterior major vertex $v$, there exists a distinct vertex in $S$ from exactly all but one of the branch paths at $v$, and this vertex is different from $v$.

Figures (3)

  • Figure 1: (a) Path (b) Tree
  • Figure 2:
  • Figure 3: Wheel $W_{n}$

Theorems & Definitions (22)

  • Definition 1
  • Definition 2
  • Definition 3: west
  • Definition 4: balajislater
  • Definition 5: balajislater
  • Theorem 6: slater
  • Lemma 7
  • proof
  • Theorem 8
  • proof
  • ...and 12 more