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DOI: 10.4230/LIPIcs.ISAAC.2025.4

Finding d-Cuts in Claw-Free Graphs

Authors: Jungho Ahn, Tala Eagling-Vose, Felicia Lucke, Daniël Paulusma, and Siani Smith

Published in: LIPIcs, Volume 359, 36th International Symposium on Algorithms and Computation (ISAAC 2025)

Abstract

The Matching Cut problem is to decide if the vertex set of a connected graph can be partitioned into two non-empty sets B and R such that the edges between B and R form a matching, that is, every vertex in B has at most one neighbour in R, and vice versa. If for some integer d ≥ 1, we allow every vertex in B to have at most d neighbours in R, and vice versa, we obtain the more general problem d-Cut. It is known that d-Cut is NP-complete for every d ≥ 1. However, for claw-free graphs, it is only known that d-Cut is polynomial-time solvable for d = 1 and NP-complete for d ≥ 3. We resolve the missing case d = 2 by proving NP-completeness. This follows from our more general study, in which we also bound the maximum degree. That is, we prove that for every d ≥ 2, d-Cut, restricted to claw-free graphs of maximum degree p, is constant-time solvable if p ≤ 2d+1 and NP-complete if p ≥ 2d+3. Moreover, in the former case, we can find a d-cut in linear time. We also show how our positive results for claw-free graphs can be generalized to S_{1^t,𝓁}-free graphs where S_{1^t,𝓁} is the graph obtained from a star on t+2 vertices by subdividing one of its edges exactly 𝓁 times.

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Jungho Ahn, Tala Eagling-Vose, Felicia Lucke, Daniël Paulusma, and Siani Smith. Finding d-Cuts in Claw-Free Graphs. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 4:1-4:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{ahn_et_al:LIPIcs.ISAAC.2025.4,
  author =	{Ahn, Jungho and Eagling-Vose, Tala and Lucke, Felicia and Paulusma, Dani\"{e}l and Smith, Siani},
  title =	{{Finding d-Cuts in Claw-Free Graphs}},
  booktitle =	{36th International Symposium on Algorithms and Computation (ISAAC 2025)},
  pages =	{4:1--4:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-408-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{359},
  editor =	{Chen, Ho-Lin and Hon, Wing-Kai and Tsai, Meng-Tsung},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2025.4},
  URN =		{urn:nbn:de:0030-drops-249121},
  doi =		{10.4230/LIPIcs.ISAAC.2025.4},
  annote =	{Keywords: matching cut, d-cut, claw-free, maximum degree}
}

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Finding d-Cuts in Claw-Free Graphs

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