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arXiv · 2608.13015

On the Restricted Edge-Cuts of Optimal 1-Planar Graphs

Licheng Zhang·Zhangdong Ouyang·Yuanqiu Huang·Guiping Wang
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Abstract

The restricted edge-connectivity of a graph is the minimum size of an edge-cut whose removal leaves every component with at least two vertices. In 2024, Zhang et al. showed that the restricted edge-connectivity of any optimal $1$-planar graph belongs to $\{8,10,12\}$. In this paper, we exclude $8$ as a possible value, thereby proving that the restricted edge-connectivity is either $10$ or $12$, and both values are attainable. Furthermore, we show that the restricted edge-connectivity of a 6-connected optimal 1-planar graph equals $10$ if and only if the graph contains an edge whose two endvertices both have degree $6$. As a key ingredient, we characterize the structure of vertex-induced subgraphs on $n$ vertices with $4n-9$ edges in optimal 1-planar graphs, and use this characterization to establish a connection between restricted edge-cuts and vertex-cuts in optimal 1-planar graphs.

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Licheng Zhang, Zhangdong Ouyang, Yuanqiu Huang, Guiping Wang. 2026-08-13. On the Restricted Edge-Cuts of Optimal 1-Planar Graphs. https://arxiv.org/abs/2608.13015

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