License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
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DOI: 10.4230/LIPIcs.IPEC.2020.8
URN: urn:nbn:de:0030-drops-133118
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Chaplick, Steven ; Golovach, Petr A. ; Hartmann, Tim A. ; Knop, Dušan

Recognizing Proper Tree-Graphs

LIPIcs-IPEC-2020-8.pdf (0.6 MB)


We investigate the parameterized complexity of the recognition problem for the proper H-graphs. The H-graphs are the intersection graphs of connected subgraphs of a subdivision of a multigraph H, and the properness means that the containment relationship between the representations of the vertices is forbidden. The class of H-graphs was introduced as a natural (parameterized) generalization of interval and circular-arc graphs by Biró, Hujter, and Tuza in 1992, and the proper H-graphs were introduced by Chaplick et al. in WADS 2019 as a generalization of proper interval and circular-arc graphs. For these graph classes, H may be seen as a structural parameter reflecting the distance of a graph to a (proper) interval graph, and as such gained attention as a structural parameter in the design of efficient algorithms. We show the following results. - For a tree T with t nodes, it can be decided in 2^{𝒪(t² log t)} ⋅ n³ time, whether an n-vertex graph G is a proper T-graph. For yes-instances, our algorithm outputs a proper T-representation. This proves that the recognition problem for proper H-graphs, where H required to be a tree, is fixed-parameter tractable when parameterized by the size of T. Previously only NP-completeness was known. - Contrasting to the first result, we prove that if H is not constrained to be a tree, then the recognition problem becomes much harder. Namely, we show that there is a multigraph H with 4 vertices and 5 edges such that it is NP-complete to decide whether G is a proper H-graph.

BibTeX - Entry

  author =	{Steven Chaplick and Petr A. Golovach and Tim A. Hartmann and Du{\v{s}}an Knop},
  title =	{{Recognizing Proper Tree-Graphs}},
  booktitle =	{15th International Symposium on Parameterized and Exact Computation (IPEC 2020)},
  pages =	{8:1--8:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-172-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{180},
  editor =	{Yixin Cao and Marcin Pilipczuk},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-133118},
  doi =		{10.4230/LIPIcs.IPEC.2020.8},
  annote =	{Keywords: intersection graphs, H-graphs, recognition, fixed-parameter tractability}

Keywords: intersection graphs, H-graphs, recognition, fixed-parameter tractability
Collection: 15th International Symposium on Parameterized and Exact Computation (IPEC 2020)
Issue Date: 2020
Date of publication: 04.12.2020

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