License: 
Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.IPEC.2020.8
URN: urn:nbn:de:0030-drops-133118
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/13311/
Chaplick, Steven ;
Golovach, Petr A. ;
Hartmann, Tim A. ;
Knop, Dušan
Recognizing Proper Tree-Graphs
Abstract
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^{?(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
@InProceedings{chaplick_et_al:LIPIcs:2020:13311,
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 = {https://drops.dagstuhl.de/opus/volltexte/2020/13311},
URN = {urn:nbn:de:0030-drops-133118},
doi = {10.4230/LIPIcs.IPEC.2020.8},
annote = {Keywords: intersection graphs, H-graphs, recognition, fixed-parameter tractability}
}
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Keywords: |
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intersection graphs, H-graphs, recognition, fixed-parameter tractability |
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Collection: |
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15th International Symposium on Parameterized and Exact Computation (IPEC 2020) |
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Issue Date: |
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2020 |
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Date of publication: |
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04.12.2020 |
