Simultaneous embeddings of graphs as median and antimedian subgraphs

The distance DG(v) of a vertex v in an undirected graph G is the sum of the distances between v and all other vertices of G. The set of vertices in G with maximum (minimum) distance is the antimedian (median) set of a graph G. It is proved that for arbitrary graphs G and J and a positive integer r ≥ 2, there exists a connected graph H such that G is the antimedian and J the median subgraphs of H , respectively, and that dH(G, J) = r. When both G and J are connected, G and J can in addition be made convex subgraphs of H .