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Kernels for the Disjoint Paths Problem on Subclasses of Chordal Graphs

Authors Juhi Chaudhary , Harmender Gahlawat , Michal Włodarczyk , Meirav Zehavi



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Juhi Chaudhary
  • Ben-Gurion University of the Negev, Beersheba, Israel
Harmender Gahlawat
  • Ben-Gurion University of the Negev, Beersheba, Israel
Michal Włodarczyk
  • University of Warsaw, Poland
Meirav Zehavi
  • Ben-Gurion University of the Negev, Beersheba, Israel

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Juhi Chaudhary, Harmender Gahlawat, Michal Włodarczyk, and Meirav Zehavi. Kernels for the Disjoint Paths Problem on Subclasses of Chordal Graphs. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 10:1-10:22, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.IPEC.2023.10

Abstract

Given an undirected graph G and a multiset of k terminal pairs 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 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 𝒪(k²) 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 NP-complete on complete graphs. Then, we design an 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 𝒪(k²) 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].

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Mathematics of computing → Graph algorithms
Keywords
  • Kernelization
  • Parameterized Complexity
  • Vertex-Disjoint Paths Problem
  • Edge-Disjoint Paths Problem

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