  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.
