Splitting and contractible edges in 4-connected graphs

Abstract

Let G be a graph and let x be a vertex of degree four with N G ( x)={ a, b, c, d}. Then the operation of deleting x and adding the edges ab and cd is called splitting at x. An edge e of a graph G is said to be k-contractible if contraction of e yields a k-connected graph. Splitting has been studied as a reduction method to preserve edge-connectivity. In this paper, we consider splitting and 4-contractible edges as tools for reduction of 4-connected graphs. We prove that for a 4-connected graph G of order at least six, there exists either a 4-contractible edge or a vertex eligible for splitting preserving 4-connectedness near every vertex in G.