On geodesic structures of weakly median graphs—II: Compactness, the role of isometric rays

We prove that the vertex set of a Kℵ0Kℵ0-free weakly median graph G   endowed with the weak topology associated with the geodesic convexity on V(G)V(G) is compact if and only if G has one of the following equivalent properties: (1) G contains no isometric rays; (2) any chain of interval of G ordered by inclusion is finite; (3) every self-contraction of G fixes a non-empty finite regular weakly median subgraph of G  . We study the self-contractions of Kℵ0Kℵ0-free weakly median graphs which fix no finite set of vertices. We also follow a suggestion of Imrich and Klavzar [Product Graphs, Wiley, New York, 2000] by defining different centers of such a graph G, each of them giving rise to a non-empty finite regular weakly median subgraph of G which is fixed by all automorphisms of G.