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Contractible Non-Edges in 3-Connected Graphs

Shuai Kou, Chengfu Qin, Weihua Yang, Mingzu Zhang

TL;DR

This work addresses the problem of classifying 3-connected graphs that contain exactly one contractible non-edge. It builds on Chan's and Kriesell's results by employing a structural framework around 3-cuts, including fragments, fans, and the notions of semi-wheels and semi-prisms, to perform a detailed case analysis. The main contribution is a complete characterization: a 3-connected graph has exactly one contractible non-edge if and only if it is isomorphic to $K_{n}^{-}$ with $n\ge 5$ or to one of four small graphs shown in Fig. 2. This result resolves the open question and clarifies how contraction preserves connectivity in highly connected graphs, with potential implications for network design and resilience analyses.

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.

Contractible Non-Edges in 3-Connected Graphs

TL;DR

This work addresses the problem of classifying 3-connected graphs that contain exactly one contractible non-edge. It builds on Chan's and Kriesell's results by employing a structural framework around 3-cuts, including fragments, fans, and the notions of semi-wheels and semi-prisms, to perform a detailed case analysis. The main contribution is a complete characterization: a 3-connected graph has exactly one contractible non-edge if and only if it is isomorphic to with or to one of four small graphs shown in Fig. 2. This result resolves the open question and clarifies how contraction preserves connectivity in highly connected graphs, with potential implications for network design and resilience analyses.

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.
Paper Structure (4 sections, 14 theorems, 2 figures)

This paper contains 4 sections, 14 theorems, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some further Lemmas
  4. Proof of Theorem \ref{['thm1']}

Key Result

Theorem A

Kriesell1 Every non-complete 3-connected finite graph neither isomorphic to a wheel nor isomorphic to one of the ten graphs in Fig. fig1 contains a contractible non-edge.

Figures (2)

  • Figure 1: 3-connected graphs other than complete graphs and wheels that do not contain any contractible non-edges
  • Figure 2: 3-connected graphs other than $K_{n}^{-}(n\geq5)$ that contain exactly one contractible non-edge $\{u, v\}$

Theorems & Definitions (25)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • ...and 15 more