Strictly Metrizable Graphs are Minor-Closed

TL;DR

We study graphs $G$ for which every consistent path system on $G$ is induced by some edge weighting as the unique $w$-shortest paths with $w:E\to \mathbb{R}_{>0}$. The main technical tool is a structural analysis using halos and persistent edges, together with a contraction-based argument that once all compliant edges are removed, strict metrizability is preserved under taking contractions, yielding minor-closure. The core structural result shows that any 2-connected strictly metrizable graph with no compliant edges is one of $K_5$, $W_5$, or a subdivision of $K_{2,3}$, $K_4$, $W_4$, or $W_4'$, providing an obstruction-like view; a zero-weight edge analysis connects persistence to feasibility of zeros and informs an inductive construction. These insights suggest a finite forbidden-minor characterization and have implications for efficient recognition of metrizability in graphs, as well as extensions to non-positive weights and related path-system realizations.

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.

Figures (10)

• Figure 1: Examples of halos in graphs.
• Figure 2: Up to adding compliant edges, all $2$-connected strictly metrizable graphs are subdivisions of one of the above graphs.
• Figure 3: The subgraph $H$ of disjoint cycles.
• Figure 4: The various scenarios of Case 2, in the proof of \ref{['prop:disjoint_cycles']}.
• Figure 5: The subcases of Case 3, in the proof of \ref{['prop:disjoint_cycles']}.

Theorems & Definitions (14)

• Theorem 1.1
• Proposition 2.1: CL
• Proposition 2.2: CL
• Lemma 2.3
• proof
• Theorem 3.1
• Proposition 3.2
• proof
• Lemma 3.3: Lovász, L
• proof