Distance Preserving Graphs

TL;DR

This paper explores the properties of distance preserving graphs, focusing on how to maintain their isometric subgraph structures when adding vertices, and identifies conditions under which graphs are or are not distance preserving.

Contribution

It introduces new conditions for adding vertices to distance preserving graphs and characterizes when certain classes of graphs are distance preserving.

Findings

Chordal graphs are distance preserving.

Graphs with large girth are not distance preserving.

Conditions for vertex addition maintaining distance preservation.

Abstract

Given a graph $G$ then a subgraph $H$ is $isometric$ if, for every pair of vertices $u,v$ of $H$, we have $d_H(u,v) = d_G(u,v)$. We say a graph $G$ is $distance\ preserving\ (dp)$ if it has an isometric subgraph of every possible order up to the order of $G$. We consider how to add a vertex to a dp graph so that the result is a dp graph. This condition implies that chordal graphs are dp. We also find a condition on the girth of $G$ which implies that it is not dp. In closing, we discuss other work and open problems concerning dp graphs.