子立方和近子立方图中的最优路径划分
Optimal Path Partitions in Subcubic and Almost-subcubic Graphs
AI总结：
研究子立方和近子立方图中边划分的最优路径问题，提出固定参数可解算法及多项式时间解法。
摘要：
我们考虑将图的边划分成尽可能少路径的问题。这是Gallai的经典猜想主题，也是组合学中的常见话题。关于最优划分图的复杂性，Peroché[Discret. Appl. Math., 1984]证明了在最大度数为四的图上，该问题已经是NP难的，即使仅询问是否只需两条路径。我们证明在子立方图上该问题可以在多项式时间内解决，然后我们为“近子立方”图提出一个高效的算法。具体来说，我们证明当参数为图到子立方图的边删除距离时，该问题是固定参数可解的。为此，我们将问题转化为在第一阶逻辑扩展了不相交路径谓词（FO+DP）中的模型检查，并然后利用Schirrmacher, Siebertz, Stamoulis, Thilikos和Vigny[LICS 2024]最近的可解性结果。
英文摘要：
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].
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