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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 thedistance sum to vertices from $\pi$. The median function satisfies severalsimple axioms: Anonymity (A), Betweeness (B), and Consistency (C). McMorris,Mulder, Novick and Powers (2015) defined the ABC-problem for consensusfunctions 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 (inparticular, bipartite Helly graphs) are ABC-graphs. On the other hand, theaddition of some simple local axioms satisfied by the median function in allgraphs (axioms (T), and (T$_2$)) enables us to show that all graphs withconnected median (comprising Helly graphs, median graphs, basis graphs ofmatroids and even $\Delta$-matroids) are ABCT-graphs and that benzenoid graphsare ABCT$_2$-graphs. McMorris et al (2015) proved that the graphs satisfyingthe pairing property (called the intersecting-interval property in their paper)are ABC-graphs. We prove that graphs with the pairing property constitute aproper subclass of bipartite Helly graphs and we discuss the complexity statusof the recognition problem of such graphs.

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