AI 中文总结
解决Dross等人关于围长至少为\(g\)的平面图反馈顶点集大小的猜想，通过将问题与带符号图联系，证明平面图反馈顶点集最小大小以上界为图所有签名的最大受挫指数。
AI 中文摘要
图的反馈顶点集是指删除后能使图成为森林的顶点集。2016年，Dross、Montassier和Pinlou猜想每个围长至少为\(g\)的平面图\(G\)都有一个大小至多为\(e(G)/g\)的反馈顶点集。在本笔记中，我们通过将此问题与带符号图联系起来证实了该猜想。带符号图\((G,\Sigma)\)的受挫指数定义为\(G\)上所有与\(\Sigma\)切换等价的签名中负边的最小数量。我们证明平面图反馈顶点集的最小大小以上界为图所有签名的最大受挫指数，从而为最小反馈顶点集的大小提供了一个紧上界，解决了Dross、Montassier和Pinlou（2016年）的猜想。
英文摘要
A feedback vertex set of a graph is a set of vertices whose deletion leaves a forest. In 2016, Dross, Montassier, and Pinlou conjectured that every planar graph $G$ of girth at least $g$ admits a feedback vertex set of size at most $e(G)/g$. In this note, we confirm this conjecture by connecting this problem with signed graphs. The frustration index of a signed graph $(G,Σ)$ is defined as the minimum number of negative edges among all signatures on $G$ that are switching-equivalent to $Σ$. Equivalently, it is the minimum number of edges whose deletion results in a balanced subgraph of $(G,Σ)$. We show that the minimum size of a feedback vertex set of a planar graph is bounded above by the maximum frustration index over all signatures of the graph, and thereby provide a tight upper bound on the size of the minimum feedback vertex set, which resolves the conjecture of Dross, Montassier, and Pinlou (2016).
