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arXiv · 2505.08185

Contractible Non-Edges in 3-Connected Graphs

Shuai Kou·Chengfu Qin·Weihua Yang·Mingzu Zhang
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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.

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Shuai Kou, Chengfu Qin, Weihua Yang, Mingzu Zhang. 2025-05-13. Contractible Non-Edges in 3-Connected Graphs. https://arxiv.org/abs/2505.08185

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