A Characterization of Uniquely Representable Graphs

TL;DR

This paper characterizes uniquely representable graphs as exactly the block graphs and explores related classes, extending the understanding of graph structures in metric and almost-metric betweenness contexts.

Contribution

It provides a complete characterization of uniquely representable graphs as block graphs and links them with distance-hereditary graphs, extending prior assumptions about trees.

Findings

Uniquely representable graphs are exactly block graphs.

Block graphs coincide with certain classes of distance-hereditary graphs.

Results extend from metric to almost-metric betweenness structures.

Abstract

The betweenness structure of a finite metric space $M = (X, d)$ is a pair $\mathcal{B}(M) = (X,\beta_M)$ where $\beta_M$ is the so-called betweenness relation of $M$ that consists of point triplets $(x, y, z)$ such that $d(x, z) = d(x, y) + d(y, z)$. The underlying graph of a betweenness structure $\mathcal{B} = (X,\beta)$ is the simple graph $G(\mathcal{B}) = (X, E)$ where the edges are pairs of distinct points with no third point between them. A connected graph $G$ is uniquely representable if there exists a unique metric betweenness structure with underlying graph $G$. It was implied by previous works that trees are uniquely representable. In this paper, we give a characterization of uniquely representable graphs by showing that they are exactly the block graphs. Further, we prove that two related classes of graphs coincide with the class of block graphs and the class of distance-hereditary graphs, respectively. We show that our results hold not only for metric but also for almost-metric betweenness structures.