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

• Authors: C. Balbuena;M. Cera;A. Diánez;P. García-Vázquez;X. Marcote
• Venue: Information Processing Letters
• Year: 2005

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 λ'(G) is the minimum cardinality over all restricted edge-cuts. A graph is said to be λ'-optimal if λ'(G) = ξ(G), where ξ(G) denotes the minimum edge-degree of G defined as ξ(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 D1 (resp. D2) 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 λ'-optimal if D1 ≤ g - 2 and D2 ≤ g - 5. For even girth we obtain a similar result. Second, let F ⊂ V(G) with |F| = δ - 1, δ ≥ 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 λ'-optimal if this Q-diameter is at most 2[(g - 3)/2].