PSEUDO-MEDIAN GRAPHS - DECOMPOSITION VIA AMALGAMATION AND CARTESIAN MULTIPLICATION

A graph is pseudo-median if for every triple u, v, w of vertices there exists either a unique vertex between each pair of them, if their mutual distances sum up to an even number, or a unique triangle whose edges lie between the three pairs of u, v, w, respectively, if the distance sum is odd. We show that every finite pseudo-median graph can be built up by successive amalgamations of smaller pieces. The building stones themselves are certain Cartesian products of wheels, snakes (i.e., path-like 2-trees), and complete graphs minus matchings.