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Edge Multiway Cut and Node Multiway Cut Are Hard for Planar Subcubic Graphs

Authors Matthew Johnson , Barnaby Martin , Sukanya Pandey , Daniël Paulusma , Siani Smith , Erik Jan van Leeuwen

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Author Details

Matthew Johnson
  • Durham University, UK
Barnaby Martin
  • Durham University, UK
Sukanya Pandey
  • Utrecht University, The Netherlands
Daniël Paulusma
  • Durham University, UK
Siani Smith
  • University of Bristol, UK
  • Heilbronn Institute for Mathematical Research, Bristol, UK
Erik Jan van Leeuwen
  • Utrecht University, The Netherlands

Acknowledgements

The authors wish to thank Jelle Oostveen for his helpful comments.

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Matthew Johnson, Barnaby Martin, Sukanya Pandey, Daniël Paulusma, Siani Smith, and Erik Jan van Leeuwen. Edge Multiway Cut and Node Multiway Cut Are Hard for Planar Subcubic Graphs. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 29:1-29:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.SWAT.2024.29

Abstract

It is known that the weighted version of Edge Multiway Cut (also known as Multiterminal Cut) is NP-complete on planar graphs of maximum degree 3. In contrast, for the unweighted version, NP-completeness is only known for planar graphs of maximum degree 11. In fact, the complexity of unweighted Edge Multiway Cut was open for graphs of maximum degree 3 for over twenty years. We prove that the unweighted version is NP-complete even for planar graphs of maximum degree 3. As weighted Edge Multiway Cut is polynomial-time solvable for graphs of maximum degree at most 2, we have now closed the complexity gap. We also prove that (unweighted) Node Multiway Cut (both with and without deletable terminals) is NP-complete for planar graphs of maximum degree 3. By combining our results with known results, we can apply two meta-classifications on graph containment from the literature. This yields full dichotomies for all three problems on H-topological-minor-free graphs and, should H be finite, on H-subgraph-free graphs as well. Previously, such dichotomies were only implied for H-minor-free graphs.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Problems, reductions and completeness
Keywords
  • multiway cut
  • planar subcubic graph
  • complexity dichotomy
  • graph containment

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