Geodeticity of the contour of chordal graphs

  • Authors:

  • José Cáceres;Carmen Hernando;Mercè Mora;Ignacio M. Pelayo;María L. Puertas;Carlos Seara

  • Affiliations:

  • Departamento de Estadística y Matemática Aplicada, Universidad de Almería, Ctra. Sacramento s/n, 04120 Almería, Spain;Departament de Matemítica Aplicada I, Universitat Politècnica de Catalunya, Avda Diagonal 647, 08028 Barcelona, Spain;Departament de Matemítica Aplicada II, Universitat Politècnica de Catalunya, Jordi Girona 1, 08034 Barcelona, Spain;Departament de Matemítica Aplicada III, Universitat Politècnica de Catalunya, Avda Canal Olímpic s/n, 08860 Castelldefels, Spain;Departamento de Estadística y Matemática Aplicada, Universidad de Almería, Ctra. Sacramento s/n, 04120 Almería, Spain;Departament de Matemítica Aplicada II, Universitat Politècnica de Catalunya, Jordi Girona 1, 08034 Barcelona, Spain

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2008

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Abstract

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. Moreover, if no vertex in the whole graph V(G) is further away from u than v, then v is called an eccentric vertex of G. A vertex v belongs to the contour of G if no neighbor of v has an eccentricity greater than the eccentricity of v. Furthermore, if no vertex in the whole graph V(G) has an eccentricity greater than the eccentricity of v, then v is called a peripheral vertex of G. This paper is devoted to study these kinds of vertices for the family of chordal graphs. Our main contributions are, firstly, obtaining a realization theorem involving the cardinalities of the periphery, the contour, the eccentric subgraph and the boundary, and secondly, proving both that the contour of every chordal graph is geodetic and that this statement is not true for every perfect graph.