On geodetic sets formed by boundary vertices

• Published in: Discrete mathematics Vol. 306; no. 2; pp. 188 - 198
• Main Authors: Cáceres, José, Hernando, Carmen, Mora, Mercè, Pelayo, Ignacio M., Puertas, María L., Seara, Carlos
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 06.02.2006; Elsevier
• Subjects: Boundary; Combinatorics; Combinatorics. Ordered structures; Contour; Eccentricity; Exact sciences and technology; Geodesic convexity; Geodetic set; Global analysis, analysis on manifolds; Graph theory; Mathematics; Periphery; Sciences and techniques of general use; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Geodetic set; Geodesic convexity; Eccentricity; Contour; Boundary; Periphery; Finite graph; Shortest path; Connected graph; Sufficient condition; Boundary vertex; Convexity; Geodesic set
• ISSN: 0012-365X; 1872-681X
• DOI: 10.1016/j.disc.2005.12.012

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 [G. Chartrand, D. Erwin, G.L. Johns, P. Zhang, Boundary vertices in graphs, Discrete Math. 263 (2003) 25–34] 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 ) ] .