A result on quasi k-connected graphs

Abstract

Let G be a k-connected graph, and T be a subset of V (G). If G-T is not connected, then T is said to be a cut-set of G. A k-cut-set T of G is a cut-set of G with |T| = k. Let T be a k-cut-set of a k-connected graph G. If G - T can be partitioned into subgraphs G ₁ and G ₂ such that |G ₁| ≥ 2, |G ₂| ≥ 2, then we call T a nontrivial k-cut-set of G. Suppose that G is a (k -1)-connected graph without nontrivial (k -1)-cut-set. Then we call G a quasi k-connected graph. In this paper, we prove that for any integer k ≥ 5, if G is a k-connected graph without K ⁻ ₄, then every vertex of G is incident with an edge whose contraction yields a quasi k-connected graph, and so there are at least $\frac{{|V(G)|}}{2}$ edges of G such that the contraction of every member of them results in a quasi k-connected graph.