Given an undirected graph $G$ and a multiset of $k$ terminal pairs $\mathcal{X}$, the Vertex-Disjoint Paths (\VDP) and Edge-Disjoint Paths (\EDP) problems ask whether $G$ has $k$ pairwise internally vertex-disjoint paths and $k$ pairwise edge-disjoint paths, respectively, connecting every terminal pair in~$\mathcal{X}$. In this paper, we study the kernelization complexity of \VDP~and~\EDP~on subclasses of chordal graphs. For \VDP, we design a $4k$ vertex kernel on split graphs and an $\mathcal{O}(k^2)$ vertex kernel on well-partitioned chordal graphs. We also show that the problem becomes polynomial-time solvable on threshold graphs. For \textsc{EDP}, we first prove that the problem is $\mathsf{NP}$-complete on complete graphs. Then, we design an $\mathcal{O}(k^{2.75})$ vertex kernel for \EDP~on split graphs, and improve it to a $7k+1$ vertex kernel on threshold graphs. Lastly, we provide an $\mathcal{O}(k^2)$ vertex kernel for \EDP~on block graphs and a $2k+1$ vertex kernel for clique paths. Our contributions improve upon several results in the literature, as well as resolve an open question by Heggernes et al.~[Theory Comput. Syst., 2015].

翻译：给定一个无向图$G$和$k$个终端对的多重集$\mathcal{X}$，顶点不交路径问题和边不交路径问题分别要求判断$G$中是否存在$k$条两两内部顶点不交的路径和$k$条两两边不交的路径，以连接集合$\mathcal{X}$中的每个终端对。本文研究了弦图子类上顶点不交路径和边不交路径问题的核化复杂性。对于顶点不交路径，我们在分裂图上设计了$4k$顶点核，并在良划分弦图上设计了$\mathcal{O}(k^2)$顶点核。我们还证明了该问题在阈值图上可在多项式时间内求解。对于边不交路径，我们首先证明该问题在完全图上为$\mathsf{NP}$-完全。然后，我们在分裂图上为边不交路径设计了$\mathcal{O}(k^{2.75})$顶点核，并在阈值图上将其改进为$7k+1$顶点核。最后，我们为块图上的边不交路径提供了$\mathcal{O}(k^2)$顶点核，并为团路径提供了$2k+1$顶点核。我们的贡献改进了文献中的若干结果，并解决了Heggernes等人[Theory Comput. Syst., 2015]提出的一个开放问题。