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].
@InProceedings{chaudhary_et_al:LIPIcs.IPEC.2023.10, author = {Chaudhary, Juhi and Gahlawat, Harmender and W{\l}odarczyk, Michal and Zehavi, Meirav}, title = {{Kernels for the Disjoint Paths Problem on Subclasses of Chordal Graphs}}, booktitle = {18th International Symposium on Parameterized and Exact Computation (IPEC 2023)}, pages = {10:1--10:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-305-8}, ISSN = {1868-8969}, year = {2023}, volume = {285}, editor = {Misra, Neeldhara and Wahlstr\"{o}m, Magnus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2023.10}, URN = {urn:nbn:de:0030-drops-194296}, doi = {10.4230/LIPIcs.IPEC.2023.10}, annote = {Keywords: Kernelization, Parameterized Complexity, Vertex-Disjoint Paths Problem, Edge-Disjoint Paths Problem} }
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