Contractible Non-Edges in 3-Connected Graphs

TL;DR

This paper characterizes all 3-connected graphs that contain exactly one contractible non-edge, extending previous work on graphs with no contractible non-edges.

Contribution

It provides a complete characterization of 3-connected graphs with exactly one contractible non-edge, solving an open problem posed by Tsz Lung Chan.

Findings

Identifies all 3-connected graphs with exactly one contractible non-edge.

Extends the classification of 3-connected graphs beyond those with no contractible non-edges.

Addresses an open problem in graph connectivity theory.

Abstract

We call a pair of non-adjacent vertices in G a non-edge. Contraction of a non-edge {u, v} in G is the replacement of u and v with a single vertex z and then making all the vertices that are adjacent to u or v adjacent to z. A non-edge {u, v} is said to be contractible in a k-connected graph G, if the resulting graph after its contraction remains k-connected. Tsz Lung Chan characterized all 3-connected graphs (finite or infinite) that does not contain any contractible non-edges in 2019, and posed the problem of characterizing all 3-connected graphs that contain exactly one contractible non-edge. In this paper, we solve this problem.