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Distance Critical Graphs

Joshua Cooper, Gabrielle Tauscheck

Abstract

In 1971, Graham and Pollak provided a formula for the determinant of the distance matrix of any tree on $n$ vertices. Yan and Yeh reproved this by exploiting the fact that pendant vertices can be deleted from trees without changing the remaining entries of the distance matrix. Considering failures of their argument to generalize invites the question: which graphs have the property that deleting any one vertex results in a change to some pairwise distance? We refer to such worst-case graphs as ``distance critical''. This work explores the structural properties of distance critical graphs, preservation of distance-criticality by products, and the nature of extremal distance critical graphs. We end with a few open questions.

Distance Critical Graphs

Abstract

In 1971, Graham and Pollak provided a formula for the determinant of the distance matrix of any tree on vertices. Yan and Yeh reproved this by exploiting the fact that pendant vertices can be deleted from trees without changing the remaining entries of the distance matrix. Considering failures of their argument to generalize invites the question: which graphs have the property that deleting any one vertex results in a change to some pairwise distance? We refer to such worst-case graphs as ``distance critical''. This work explores the structural properties of distance critical graphs, preservation of distance-criticality by products, and the nature of extremal distance critical graphs. We end with a few open questions.
Paper Structure (6 sections, 22 theorems, 3 equations, 4 figures, 2 tables)

This paper contains 6 sections, 22 theorems, 3 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Structural Properties
  4. Graph Products
  5. Extremal Results
  6. Open Questions

Key Result

Proposition 2.4

A connected graph $G$ is distance critical if and only if for all $v \in V(G)$, $v$ admits a determining pair $\{a,b\}$ where $a,b \in V(G)$.

Figures (4)

  • Figure 1: Dodecahedron
  • Figure 2: Maximal distance critical graph on $8$ vertices containing $C_8$ as a proper subgraph.
  • Figure 3: Maximum degree construction for $n=12$.
  • Figure 4: $\Gamma_5$, with $A$ vertices white, $B$ vertices gray, and $C$ vertices black.

Theorems & Definitions (49)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Corollary 2.5
  • Corollary 2.6
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 39 more