Kernels for the Disjoint Paths Problem on Subclasses of Chordal Graphs

TL;DR

This paper investigates kernelization algorithms for the Vertex-Disjoint Paths and Edge-Disjoint Paths problems on various subclasses of chordal graphs, providing new kernels and complexity results.

Contribution

It introduces new polynomial kernels for extsc{VDP} and extsc{EDP} on subclasses of chordal graphs, and resolves an open problem regarding the complexity on complete graphs.

Findings

Polynomial kernels for extsc{VDP} on split and well-partitioned chordal graphs.

NP-completeness of extsc{EDP} on complete graphs.

Improved kernel sizes for extsc{EDP} on split, threshold, block graphs, and clique paths.

Abstract

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].