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

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.