Sufficient conditions for λ'-optimality of graphs with small conditional diameter

  • Authors:

  • C. Balbuena;M. Cera;A. Diánez;P. García-Vázquez;X. Marcote

  • Affiliations:

  • Departament de Matemática Aplicada III, Universitat Politècnica de Catalunya, Barcelona, Spain;Departamento de Matemática Aplicada I, Universidad de Sevilla, Sevilla, Spain;Departamento de Matemática Aplicada I, Universidad de Sevilla, Sevilla, Spain;Departamento de Matemática Aplicada I, Universidad de Sevilla, Sevilla, Spain;Departament de Matemática Aplicada III, Universitat Politècnica de Catalunya, Barcelona, Spain

  • Venue:
  • Information Processing Letters
  • Year:
  • 2005

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Abstract

A restricted edge-cut S of a connected graph G is an edge-cut such that G-S has no isolated vertex. The restricted edge-connectivity @l^'(G) is the minimum cardinality over all restricted edge-cuts. A graph is said to be @l^'-optimal if @l^'(G)=@x(G), where @x(G) denotes the minimum edge-degree of G defined as @x(G)=min{d(u)+d(v)-2:uv@?E(G)}. The P-diameter of G measures how far apart a pair of subgraphs satisfying a given property P can be, and hence it generalizes the standard concept of diameter. In this paper we prove two kind of results, according to which property P is chosen. First, let D"1 (resp. D"2) be the P-diameter where P is the property that the corresponding subgraphs have minimum degree at least one (resp. two). We prove that a graph with odd girth g is @l^'-optimal if D"1==2, being the minimum degree of G. Using the property Q of being vertices of G-F we prove that a graph with girth g@?{4,6,8} is @l^'-optimal if this Q-diameter is at most 2@?(g-3)/2@?.