Recognizing Proper Tree-Graphs

Steven Chaplick, Petr A. Golovach, Tim A. Hartmann, Dusan Knop

Research output: Chapter in Book/Report/Conference proceedingChapterAcademic


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( t 2 log t ) · n 3 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.

Original languageEnglish
Title of host publication15th International Symposium on Parameterized and Exact Computation (IPEC 2020)
EditorsY. Cao, M. Pilipczuk
Place of PublicationDagstuhl, Germany
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Print)978-3-95977-172-6
Publication statusPublished - 2020
Event15th International Symposium on Parameterized and Exact Computation - Online, Hongkong, China
Duration: 14 Dec 202018 Dec 2020
Conference number: 15

Publication series

SeriesLeibniz International Proceedings in Informatics


Symposium15th International Symposium on Parameterized and Exact Computation
Abbreviated titleIPEC 2020
Internet address

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