2026-07-26T17:26:45Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:13311 2024-03-06T10:51:40Z ddc:004 open_access

Recognizing Proper Tree-Graphs Chaplick, Steven Golovach, Petr A. Hartmann, Tim A. Knop, Dušan intersection graphs H-graphs recognition fixed-parameter tractability 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Steven Chaplick and Petr A. Golovach and Tim A. Hartmann and Dušan Knop 2020 Is Part Of LIPIcs, Volume 180, 15th International Symposium on Parameterized and Exact Computation (IPEC 2020) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.IPEC.2020.8 urn:nbn:de:0030-drops-133118 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2020.8 eng https://creativecommons.org/licenses/by/3.0/legalcode