Complexity Framework for Forbidden Subgraphs II: Edge Subdivision and the “H”-Graphs

Vadim Lozin, Barnaby Martin, Sukanya Pandey, Daniël Paulusma, Mark Siggers, Siani Smith, Erik Jan van Leeuwen

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

1 Scopus citations

Abstract

For a fixed set H of graphs, a graph G is H-subgraph-free if G does not contain any H ∈ H as a (not necessarily induced) subgraph. A recent framework gives a complete classification on H-subgraph-free graphs (for finite sets H) for problems that are solvable in polynomial time on graph classes of bounded treewidth, NP-complete on subcubic graphs, and whose NP-hardness is preserved under edge subdivision. While a lot of problems satisfy these conditions, there are also many problems that do not satisfy all three conditions and for which the complexity in H-subgraph-free graphs is unknown. We study problems for which only the first two conditions of the framework hold (they are solvable in polynomial time on classes of bounded treewidth and NP-complete on subcubic graphs, but NP-hardness is not preserved under edge subdivision). In particular, we make inroads into the classification of the complexity of four such problems: Hamilton Cycle, k-Induced Disjoint Paths, C5-Colouring and Star 3-Colouring. Although we do not complete the classifications, we show that the boundary between polynomial time and NP-complete differs among our problems and also from problems that do satisfy all three conditions of the framework, in particular when we forbid certain subdivisions of the “H”-graph (the graph that looks like the letter “H”). Hence, we exhibit a rich complexity landscape among problems for H-subgraph-free graph classes.

Original languageEnglish
Title of host publication35th International Symposium on Algorithms and Computation, ISAAC 2024
EditorsJulian Mestre, Anthony Wirth
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959773546
DOIs
StatePublished - 4 Dec 2024
Event35th International Symposium on Algorithms and Computation, ISAAC 2024 - Sydney, Australia
Duration: 8 Dec 202411 Dec 2024

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume322
ISSN (Print)1868-8969

Conference

Conference35th International Symposium on Algorithms and Computation, ISAAC 2024
Country/TerritoryAustralia
CitySydney
Period8/12/2411/12/24

Keywords

  • complexity dichotomy
  • edge subdivision
  • forbidden subgraph
  • treewidth

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