Contractible Non-Edges in 3-Connected Graphs
Combinatorics
2025-05-14 v1
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.
Cite
@article{arxiv.2505.08185,
title = {Contractible Non-Edges in 3-Connected Graphs},
author = {Shuai Kou and Chengfu Qin and Weihua Yang and Mingzu Zhang},
journal= {arXiv preprint arXiv:2505.08185},
year = {2025}
}