Disconnected Cuts in Claw-free Graphs

Martin, B.; Paulusma, D.; van Leeuwen, E. J.

Abstract

A disconnected cut of a connected graph is a vertex cut that itself also induces a disconnected subgraph. The corresponding decision problem is called Disconnected Cut. It is known that Disconnected Cut is NP-hard on general graphs, while polynomial-time algorithms exist for several graph classes. However, the complexity of the problem on claw-free graphs remained an open question. Its connection to the complexity of the problem to contract a claw-free graph to the 4-vertex cycle C4 led Ito et al. (TCS 2011) to explicitly ask to resolve this open question. We prove that Disconnected Cut is polynomial-time solvable on claw-free graphs, answering the question of Ito et al. The basis for our result is a decomposition theorem for claw-free graphs of diameter 2, which we believe is of independent interest and builds on the research line initiated by Chudnovsky and Seymour (JCTB 2007–2012) and Hermelin et al. (ICALP 2011). On our way to exploit this decomposition theorem, we characterize how disconnected cuts interact with certain cobipartite subgraphs, and prove two further algorithmic results, namely that Disconnected Cut is polynomial-time solvable on circular-arc graphs and line graphs.

Citation

Martin, B., Paulusma, D., & van Leeuwen, E. J. (2018, August). Disconnected Cuts in Claw-free Graphs. Presented at 26th Annual European Symposium on Algorithms (ESA 2018)., Helsinki, Finland

• Presentation Conference Type: Conference Paper (published)
• Conference Name: 26th Annual European Symposium on Algorithms (ESA 2018).
• Start Date: Aug 20, 2018
• End Date: Aug 24, 2018
• Acceptance Date: Jun 15, 2018
• Publication Date: Jan 1, 2018
• Deposit Date: Jul 12, 2018
• Publicly Available Date: Jul 13, 2018
• Volume: 112
• Pages: 61:1-61:14
• Series Title: Leibniz International Proceedings in Informatics (LIPIcs)
• Series ISSN: 1868-8969
• Book Title: 26th Annual European Symposium on Algorithms (ESA 2018).
• DOI: https://doi.org/10.4230/lipics.esa.2018.61
• Public URL: https://durham-repository.worktribe.com/output/1144284