On the contour of graphs

  • Authors:

  • D. Artigas;S. Dantas;M. C. Dourado;J. L. Szwarcfiter;S. Yamaguchi

  • Affiliations:

  • Instituto de Ciência e Tecnologia, Universidade Federal Fluminense, Brazil;Instituto de Matemática e Estatística, Universidade Federal Fluminense, Brazil;Instituto de Matemática, Universidade Federal do Rio de Janeiro, Brazil;COPPE-Sistemas, Universidade Federal do Rio de Janeiro, Brazil and Instituto de Matemática, Universidade Federal do Rio de Janeiro, Brazil and NCE, Universidade Federal do Rio de Janeiro, Bra ...;Shonan Institute of Technology, Fujisawa, Kanagawa 251-8511, Japan

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2013

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Abstract

Let G=(V,E) be a finite, simple and connected graph. Let S@?V, its closed interval I[S] is the set of all vertices lying on a shortest path between any pair of vertices of S. The set S is geodetic if I[S]=V. The eccentricity of a vertex v is the number of edges in the greatest shortest path between v and any vertex w of G. The contour Ct(G) of G is the set formed by vertices v such that no neighbor of v has an eccentricity greater than v. We consider the problem of determining whether the contour of a graph class is geodetic. The diameter diam(G) of G is the maximum eccentricity of the vertices in V. In this work we establish a relation between the diameter and the geodeticity of the contour of a graph. We prove that the contour is geodetic for graphs with diameter k@?4. Furthermore, for every k4, there is a graph with diameter k and whose contour is not geodetic. We show that the contour is geodetic for bipartite graphs with diameter k@?7, and for any k7 there is a bipartite graph with diameter k and non-geodetic contour. By applying these results, we solve the open problems mentioned by Caceres et al. (2008, 2005) [4,5] namely to decide whether the contour of cochordal graph, parity graph and bipartite graphs are geodetic.