Graphs with four boundary vertices

Tobias Müller (1) , Attila Pór (2) , Jean-Sébastien Sereni (3, 4)

1 CWI - Centrum voor Wiskunde en Informatica
2 WKU - Department of Mathematics - Western Kentucky University
3 LIAFA - Laboratoire d'informatique Algorithmique : Fondements et Applications
4 KAM - Department of Applied Mathematics (KAM)

A vertex v of a graph G is a boundary vertex if there exists a vertex u such that the distance in G from u to v is at least the distance from u to any neighbour of v. We give a full description of all graphs that have exactly four boundary vertices, which answers a question of Hasegawa and Saito. To this end, we introduce the concept of frame of a graph. It allows us to construct, for every positive integer b and every possible ``distance-vector'' between b points, a graph G with exactly b boundary vertices such that every graph with b boundary vertices and the same distance-vector between them is an induced subgraph of G.

Domaines: Mathématique discrète [cs.DM]

Tobias Müller, Attila Pór, Jean-Sébastien Sereni. Graphs with four boundary vertices. The Electronic Journal of Combinatorics, 2011, 18 (1), pp.P11. ⟨hal-00558913⟩