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Recognizing Proper Tree-Graphs

Authors Steven Chaplick , Petr A. Golovach , Tim A. Hartmann , Dušan Knop

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Steven Chaplick
  • Maastricht University, The Netherlands
Petr A. Golovach
  • Department of Informatics, University of Bergen, Norway
Tim A. Hartmann
  • RWTH Aachen, Germany
Dušan Knop
  • Department of Theoretical Computer Science, Faculty of Information Technology, Czech Technical University in Prague, Czech Republic


We thank Peter Zeman for initial discussions on this work, particularly relating to the NP-completeness of proper H-graph recognition. We also thank an anonymous reviewer to point to a mistake in an earlier version.

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Steven Chaplick, Petr A. Golovach, Tim A. Hartmann, and Dušan Knop. Recognizing Proper Tree-Graphs. In 15th International Symposium on Parameterized and Exact Computation (IPEC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 180, pp. 8:1-8:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Theory of computation → Fixed parameter tractability
  • intersection graphs
  • H-graphs
  • recognition
  • fixed-parameter tractability


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