On geodetic sets formed by boundary vertices
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 showin...
| Authors: | , , , , , |
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| Format: | article |
| Publication Date: | 2003 |
| Country: | España |
| Institution: | Universitat Politècnica de Catalunya (UPC) |
| Repository: | UPCommons. Portal del coneixement obert de la UPC |
| Language: | English |
| OAI Identifier: | oai:upcommons.upc.edu:2117/923 |
| Online Access: | https://hdl.handle.net/2117/923 |
| Access Level: | Open access |
| Keyword: | 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 |
| Summary: | 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)]. |
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