Kernels for the Disjoint Paths Problem on Subclasses of Chordal Graphs

Given an undirected graph G and a multiset of k terminal pairs $X$, the Vertex-Disjoint Paths (

) and Edge-Disjoint Paths (

) problems ask whether G has k pairwise internally vertex-disjoint paths and k pairwise edge-disjoint paths, respectively, connecting every terminal pair in $X$. In this paper, we study the kernelization complexity of

and

on subclasses of chordal graphs. For

, we design a 4k vertex kernel on split graphs and an $O\left(k^2\right)$ vertex kernel on well-partitioned chordal graphs. We also show that the problem becomes polynomial-time solvable on threshold graphs. For EDP, we first prove that the problem is $\mathrm{NP}$-complete on complete graphs. Then, we design an $O\left(k^{2.75}\right)$ vertex kernel for

on split graphs, and improve it to a $7k+1$ vertex kernel on threshold graphs. Lastly, we provide an $O\left(k^2\right)$ vertex kernel for

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. (2015) [27].