A set of vertices of a graph $G$ is said to be decycling if its removal leaves an acyclic subgraph. The size of a smallest decycling set is the decycling number of $G$. Generally, at least $\lceil(n+2)/4\rceil$ vertices have to be removed in order to decycle a cubic graph on $n$ vertices. In 1979, Payan and Sakarovitch proved that the decycling number of a cyclically $4$-edge-connected cubic graph of order $n$ equals $\lceil (n+2)/4\rceil$. In addition, they characterised the structure of minimum decycling sets and their complements. If $n\equiv 2\pmod4$, then $G$ has a decycling set which is independent and its complement induces a tree. If $n\equiv 0\pmod4$, then one of two possibilities occurs: either $G$ has an independent decycling set whose complement induces a forest of two trees, or the decycling set is near-independent (which means that it induces a single edge) and its complement induces a tree. In this paper we strengthen the result of Payan and Sakarovitch by proving that the latter possibility (a near-independent set and a tree) can always be guaranteed. Moreover, we relax the assumption of cyclic $4$-edge-connectivity to a significantly weaker condition expressed through the canonical decomposition of 3-connected cubic graphs into cyclically $4$-edge-connected ones. Our methods substantially use a surprising and seemingly distant relationship between the decycling number and the maximum genus of a cubic graph.

翻译：图$G$的顶点集被称为解除循环集，若移除该集合后得到无圈子图。最小解除循环集的大小称为$G$的解除循环数。一般而言，移除至少$\lceil(n+2)/4\rceil$个顶点才能解除一个$n$阶三次图的循环。1979年，Payan和Sakarovitch证明了：对于$n$阶循环$4$-边连通三次图，其解除循环数等于$\lceil (n+2)/4\rceil$。此外，他们刻画了最小解除循环集及其补集的结构。若$n\equiv 2\pmod4$，则$G$存在一个独立解除循环集，其补集诱导一棵树。若$n\equiv 0\pmod4$，则出现两种可能之一：要么$G$存在独立解除循环集，其补集诱导一个由两棵树构成的森林；要么解除循环集是近独立的（即它仅诱导一条边），且其补集诱导一棵树。本文通过证明后一种可能性（近独立集与一棵树）始终可被保证，从而强化了Payan和Sakarovitch的结果。此外，我们将循环$4$-边连通性假设放宽为一个通过将3连通三次图规范分解为循环$4$-边连通图所表达的显著更弱条件。我们的方法实质性地利用了解除循环数与三次图最大亏格之间看似遥远的惊人关联。