On subgraphs with prescribed eccentricities

A well-known result by Hedetniemi states that for every graph G there is a graph H whose center is G. We extend this result by showing under which conditions there exists, for a given graph G in which each vertex v has an integer label ℓ (v), a graph H containing G as an induced subgraph such that the eccentricity, in H, of every vertex v of G equals ℓ(v). Such a labelled graph G is said to be eccentric, and strictly eccentric if there exists such a graph H such that no vertex of H -G has the same eccentricity in H as any vertex of G. We find necessary and sufficient conditions for a labelled graph to be eccentric and for a forest to be eccentric or strictly eccentric in a tree.