Connected graphs with prescribed median and periphery

The eccentricity e(v) of a vertex v in a connected graph G is the distance between v and a vertex furthest from v in G. The center C(G) of G is the subgraph induced by those vertices of G having minimum eccentricity; the periphery P(G) is the subgraph induced by those vertices of G having maximum eccentricity. The distance d(v) of a vertex v in G is the sum of the distances from v to the vertices of G. The median M(G) of G is the subgraph induced by those vertices having minimum distance. For graphs F and G and a positive integer m, necessary and sufficient conditions are given for F and G to be the median and periphery, respectively, of some connected graph such that the distance between the median and periphery is m. Necessary and sufficient conditions are also given for two graphs to be the median and periphery and to intersect in any common induced subgraph.