Strictly Metrizable Graphs are Minor-Closed

TL;DR

This paper characterizes strictly metrizable graphs, showing they are minor-closed, by analyzing consistent path systems and their realization as geodesics under positive edge weights.

Contribution

It proves that the class of strictly metrizable graphs is minor-closed, linking path system consistency with graph minor theory.

Findings

Strictly metrizable graphs form a minor-closed family.

Every consistent path system in such graphs can be realized as geodesics.

The paper establishes a structural property of these graphs.

Abstract

A consistent path system in a graph $G$ is an collection of paths, with exactly one path between any two vertices in $G$. A path system is said to be consistent if it is intersection-closed. We say that $G$ is strictly metrizable if every consistent path system in $G$ can be realized as the system of unique geodesics with respect to some assignment of positive edge weight. In this paper, we show that the family of strictly metrizable graphs is minor-closed.