A Constructive Characterization of 4-Connected Graphs

Abstract

Let k be an integer such that $k\ge 3.$ Let ${\mathscr{H}}_{k}$ denote the set of k-connected graphs each G of which has a vertex x such that $G-x$ is a $(k-1)$-connected $(k-1)$-regular graph. Note that ${\mathscr{H}}_{3}$ is the set of wheels. Let G be a k-connected graph where $k\ge 2$ is an integer. An operation on G is defined as follows. (I) Delete a vertex x of degree at least $2(k-1)$ from G,  (II) Add new two vertices $x_{1}$ and $x_{2},$ (III) Join $x_{i}$ to $N_{i}\cup \{x_{3-i}\}$ for $i=1, 2$ where $N_{G}(x)=N_{1}\cup N_{2},$ $|N_{i}|\ge k-1$ for $i=1,2$ and $N_{1}\cap N_{2}=\emptyset .$ We call this operation “proper vertex-splitting”. Two edges of a graph are said to be “independent” if they have no common end vertex and a set of edges is said to be “independent” if each two of it are independent. Let G be a 2k-connected graph where k is a positive integer. We define an operation on G as follows. (I) Choose independent k edges of G,  (II) Subdivide each of the chosen k edges by one vertex, (III) Identify the new k vertices arising from the subdivisions. We call this operation “edge-binding”.

Tutte gave a constructive characterization of 3-connected graphs.

Theorem (Tutte’s wheel theorem, 1961) Every 3-connected graph can be obtained from a graph in ${\mathscr{H}}_{3}$ by repeated applications of edge addings and proper vertex-splittings.

In this paper we prove the following 4-connected analogue of Tutte’s Wheel Theorem.

Theorem Every 4-connected graph can be obtained from a graph in ${\mathscr{H}}_{4}$ by repeated applications of edge addings, proper vertex-splittings and edge-bindings.