ABC(T)-graphs: an axiomatic characterization of the median procedure in graphs with connected and G$^2$-connected medians

TL;DR

This paper characterizes graphs where the median function uniquely satisfies certain axioms, identifying classes like modular and Helly graphs as ABC-graphs, and explores the properties and recognition complexity of these graph classes.

Contribution

It provides an axiomatic characterization of median functions in graphs, identifying new classes of ABC-graphs and extending understanding of median-related graph properties.

Findings

Modular graphs with G^2-connected medians are ABC-graphs.

Graphs with connected medians include Helly, median, and matroid basis graphs.

Benzenoid graphs are identified as ABCT_2-graphs.

Abstract

The median function is a location/consensus function that maps any profile $\pi$ (a finite multiset of vertices) to the set of vertices that minimize the distance sum to vertices from $\pi$. The median function satisfies several simple axioms: Anonymity (A), Betweeness (B), and Consistency (C). McMorris, Mulder, Novick and Powers (2015) defined the ABC-problem for consensus functions on graphs as the problem of characterizing the graphs (called, ABC-graphs) for which the unique consensus function satisfying the axioms (A), (B), and (C) is the median function. In this paper, we show that modular graphs with $G^2$-connected medians (in particular, bipartite Helly graphs) are ABC-graphs. On the other hand, the addition of some simple local axioms satisfied by the median function in all graphs (axioms (T), and (T$_2$)) enables us to show that all graphs with connected median (comprising Helly graphs, median graphs, basis graphs of matroids and even $\Delta$-matroids) are ABCT-graphs and that benzenoid graphs are ABCT$_2$-graphs. McMorris et al (2015) proved that the graphs satisfying the pairing property (called the intersecting-interval property in their paper) are ABC-graphs. We prove that graphs with the pairing property constitute a proper subclass of bipartite Helly graphs and we discuss the complexity status of the recognition problem of such graphs.