2026-04-13T04:57:47Zhttps://recercat.cat/oai/request

oai:recercat.cat:2117/9232025-07-17T03:04:05Zcom_2072_1033col_2072_452950

On geodetic sets formed by boundary vertices Cáceres González, José Hernando Martín, María del Carmen Mora Giné, Mercè Pelayo Melero, Ignacio Manuel Puertas González, María Luz Seara Ojea, Carlos Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada I Universitat Politècnica de Catalunya. EGSA - Equacions Diferencials, Geometria, Sistemes Dinàmics i de Control, i Aplicacions Graph theory Convex geometry Boundary contour eccentricity geodesic convexity geodetic set periphery Grafs, Teoria de Geometria convexa Classificació AMS::05 Combinatorics::05C Graph theory Classificació AMS::52 Convex and discrete geometry::52A General convexity Let G be a finite simple connected graph. A vertex v is a boundary vertex of 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. Before showing that one of the main results in [3] does not hold for one of the cases, we establish a realization theorem that not only corrects the mentioned wrong statement but also improves it. Given S ⊆ V (G), its geodetic closure I[S] is the set of all vertices lying on some shortest path joining two vertices of S. We prove that the boundary vertex set ∂(G) of any graph G is geodetic, that is, I[∂(G)] = V (G). A vertex v belongs to the contour Ct(G) of G if no neighbor of v has an eccentricity greater than v. We present some sufficient conditions to guarantee the geodeticity of either the contour Ct(G) or its geodetic closure I[Ct(G)]. 2003 Article https://hdl.handle.net/2117/923 eng http://creativecommons.org/licenses/by-nc-nd/2.5/es/ Open Access Attribution-NonCommercial-NoDerivs 2.5 Spain 16 application/postscript