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

• Published in: Discrete Applied Mathematics 2024, Vol.359, p.55-74
• Main Authors: Bénéteau, Laurine, Chalopin, Jérémie, Chepoi, Victor, Vaxès, Yann
• Format: Article
• Language: English
• Subjects: Combinatorics; Computational Geometry; Computer Science; Data Structures and Algorithms; Discrete Mathematics; Mathematics; Metric Geometry
• ISSN: 0166-218X
• DOI: 10.1016/j.dam.2024.07.023

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.