We consider the problem of partitioning the edges of a graph into as few paths as possible. This is a~subject of the classic conjecture of Gallai and a recurring topic in combinatorics. Regarding the complexity of partitioning a graph optimally, Peroché [Discret. Appl. Math., 1984] proved that it is NP-hard already on graphs of maximum degree four, even when we only ask if two paths suffice. We show that the problem is solvable in polynomial time on subcubic graphs and then we present an efficient algorithm for ``almost-subcubic'' graphs. Precisely, we prove that the problem is fixed-parameter tractable when parameterized by the edge-deletion distance to a subcubic graph. To this end, we reduce the task to model checking in first-order logic extended by disjoint-paths predicates ($\mathsf{FO}\text{+}\mathsf{DP}$) and then we employ the recent tractability result by Schirrmacher, Siebertz, Stamoulis, Thilikos, and Vigny [LICS 2024].

翻译：我们考虑将图边划分为尽可能少路径的问题。这是Gallai经典猜想的研究主题，也是组合数学中反复出现的课题。关于图最优划分的复杂度问题，Peroché [Discret. Appl. Math., 1984] 证明了该问题在最大度数为四的图上已是NP困难问题，即使仅需判断两条路径是否足够。我们证明该问题在次三次图上存在多项式时间解法，并提出适用于"近次三次"图的高效算法。具体而言，我们证明该问题在参数化为到次三次图的边删除距离时是固定参数可处理的。为此，我们将任务归约为带不相交通路谓词的一阶逻辑模型检测（$\mathsf{FO}\text{+}\mathsf{DP}$），并运用Schirrmacher、Siebertz、Stamoulis、Thilikos和Vigny [LICS 2024] 近期提出的可处理性结果。