Reduction for 3-Connected Graphs of Minimum Degree at Least Four

Abstract

We give a reduction theorem for 3-connected graphs of minimum degree at least four. Let ${\mathcal{P}}_{3,4}$ be the class of 3-connected graphs of minimum degree at least four. For a vertex x of degree four in G, splitting at x is an operation of adding at most two independent edges connecting some of the neighbors of x and deleting x. A set of five vertices C = {a, b, x, y, z} in G is said to be a crown if N _G (a) = {b, x, y, z} and N _G (b) = {a, x, y, z}. Given a crown C in G, reduction of C is the operation of deleting {a, b} and adding all the missing edges among {x, y, z}. In this paper, we prove that for every vertex x in a graph $G\in{\mathcal{P}}_{3,4}$ , there exists either (1) an edge e such that its contraction yields a graph in ${\mathcal{P}}_{3,4}$ and at least one of its endvertices is of distance at most two from x, (2) a vertex v of degree four such that v is of distance at most two from x and some splitting at v yields a graph in ${\mathcal{P}}_{3,4}$ , or (3) a crown C containing x whose reduction yields a graph in ${\mathcal{P}}_{3,4}$ .