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

TL;DR

The work advances axiomatic characterizations of the median in graphs by establishing ABCT- and ABC-graph classifications for broad graph families. It shows that graphs with connected medians are ABCT-graphs and modular graphs with $G^2$-connected medians are ABC-graphs, with bipartite Helly graphs and benzenoids occupying key roles. A refined pairing framework distinguishes when ABC-functions coincide with medians, including local-to-global criteria and co-NP containment for double-pairing recognition. It also introduces new axioms $(T_2)$ and $(E_k)$ to handle equilateral metric triangles and demonstrates that benzenoids are ABCT$_2$- and ABCE$_2$-graphs, though not ABC-graphs in general. The results unify several classical graph classes under the ABC/ABCT paradigm and open questions about the exact boundary and complexity of recognition for these properties, with potential implications for consensus in networked systems.

Abstract

The median function is a location/consensus function that maps any profile $π$ (a finite multiset of vertices) to the set of vertices that minimize the distance sum to vertices from $π$. 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 $Δ$-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.

Figures (2)

• Figure 1: Triangle and quadrangle conditions
• Figure 2: The graphs from the proof of Proposition \ref{['bar->notpairing']}

Theorems & Definitions (95)

• Theorem 1: GpConMed
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• Lemma 6
• proof