Geodesic
On subgraphs with prescribed eccentricities
Discussiones Mathematicae. Graph Theory, Tome 43 (2023) no. 3, pp. 685-702

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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.
Keywords: distance, eccentricity, subgraph, tree 
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Dankelmann, Peter; DeVilbiss, Matthew; Erwin, David J.; Guest, Kelly; Matzke, Ryan. On subgraphs with prescribed eccentricities. Discussiones Mathematicae. Graph Theory, Tome 43 (2023) no. 3, pp. 685-702. http://geodesic.mathdoc.fr/item/DMGT_2023_43_3_a6/