# 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 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.

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