The median function on median graphs and semilattices (Q1975374)

Let \(\pi = (x_1, x_2, \dots, x_k)\) be a \(k\)-tuple of vertices in a finite connected graph \(G\). A median of \(\pi\) is a vertex \(x\) for which \(\sum^k_{i=1} d(x, x_i)\) is minimum, where \(d\) is a geodesic metric on \(G\). The median function on \(G\) is the function \(M\), defined by \(M(\pi) = \{x |x \text{ is a median of } \pi\}\), whose domain is the set of all \(k\)-tuples, \(k > 0\). The authors give a new characterization of the median function of a median graph \(G\) and use it to give a characterization of the median function on median semilattices.