Pseudo-median graphs are join spaces

Bandelt, H.-J. and H.M. Mulder, Pseudo-median graphs are join spaces, Discrete Mathematics 109 (1992) 13-26. Median graphs are known to share a number of geometrica! properties with Euclidean space, as is testified by the fact that median graphs ale (c!osed) join spaces-the segments of the associated join space being the geodesic intervals of the graph. As is shown in this paper the median join space model extends to a larger class of graphs, viz.: the pseudo-median graphs. A pseudo-median graph possesses for each triple U, v, w of vertices either a unique median vertex (lying geodesically between each pair of u, v, w) or a unique pseudo-median triangle (the edges of which are on geodesics between two of N, v, w each). The problem of characterizing all graphs giving rise to join spaces remains open.