Distance Critical Graphs

TL;DR

This paper investigates the structural properties of 'distance critical' graphs, which are graphs where removing any vertex alters some pairwise distances, exploring their characteristics, how they behave under graph products, and identifying extremal cases.

Contribution

It introduces the concept of distance critical graphs, analyzes their structural properties, and examines their behavior under graph operations and extremal conditions.

Findings

Identification of structural properties of distance critical graphs

Analysis of distance criticality preservation under graph products

Characterization of extremal distance critical graphs

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.