最优1-平面图的受限边割
On the Restricted Edge-Cuts of Optimal 1-Planar Graphs
AI总结：
本文排除最优1-平面图受限边连通度为8的可能，证明其为10或12且均可达，还明确6连通最优1-平面图受限边连通度为10的充要条件，刻画相关顶点诱导子图结构并建立受限边割与顶点割的联系。
AI中文摘要：
图的受限边连通度是指移除某边割后，使得每个连通分量至少包含2个顶点的边割的最小规模。2024年，Zhang等人证明了任意最优1-平面图的受限边连通度属于{8,10,12}。本文排除了8作为可能值，从而证明该受限边连通度为10或12，且两个值均可达到。此外，我们证明6连通最优1-平面图的受限边连通度等于10当且仅当该图存在一条边，其两个端点的度数均为6。作为关键要素，我们刻画了最优1-平面图中具有n个顶点、4n-9条边的顶点诱导子图的结构，并利用该刻画建立了最优1-平面图中受限边割与顶点割之间的联系。
英文摘要：
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.
