From copair hypergraphs to median graphs with latent vertices

The purpose of this paper is to extend the Buneman construction of partially labelled trees to the general case. This extension is related with the characterization of median graphs by Mulder and Schrijver. In the first section, we construct a graph G(H) associated with a copair hypergraph H on a finite set X and define the notion of a median graph with latent vertices (called X-median graph). The latent vertices (i.e. the vertices who are not labelled by elements of X) are obtained by iterating the median operation from actual (labelled) vertices. In the second section, we prove that the graph G(H) is an X-median graph. Then, in the last section, we study some special cases, the Buneman result is reobtained and the hypergraphs whose associated graphs are Hasse diagrams of distributive lattices are characterized.