Searching for geodetic boundary vertex sets

A vertex v is a boundary vertex of a connected graph G if there exists a vertex u such that no neighbor of v is further away from u than v. We obtain a number of properties involving different types of boundary vertices: peripheral, contour and eccentric vertices, including a realization theorem that not only corrects a wrong statement detected in [2], but also improves it. We also prove that the boundary vertex set ∂(G) of any graph G is geodetic, that is, every vertex in G lies on some shortest path joining two boundary vertices. A vertex v belongs to the contour Ct(G) of G if no neighbor of v has an eccentricity greater than those of v. We study the geodeticity of the contour Ct(G) and other related sets.