Constructing uniform central graphs and embedding into them

Abstract

A graph is called uniform central (UC) if all its central vertices have the same set of eccentric vertices. It is proved that if G is a UC graph with radius at least 3, then substituting a central vertex u of G with an arbitrary graph H and connecting the vertices of H to all neighbors of u (in G), yields a UC graph again. This construction extends several earlier ones and enables a simple argument for the fact that for any r ≥ 2 and any r + 1 ≤ d ≤ 2r, there exists a non-trivial UC graph G with rad(G) = r and diam(G) = d. Embeddings of graphs into UC graphs are also considered. It is shown that if G is an arbitrary graph with at least one edge then at most three additional vertices suffice to embed G into an r-UC graph with r ≥ 2. It is also proved that P3 is the only UC graph among almost self-centered graphs.