On H-Topological Intersection Graphs

Steven Chaplick, Martin Topfer, Jan Vobornik, Peter Zeman*

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review


Biro et al. (Discrete. Math 100(1-3):267-279, 1992) introduced the concept of H-graphs, intersection graphs of connected subgraphs of a subdivision of a graph H. They are related to and generalize many important classes of geometric intersection graphs, e.g., interval graphs, circular-arc graphs, split graphs, and chordal graphs. Our paper starts a new line of research in the area of geometric intersection graphs by studying several classical computational problems on H-graphs: recognition, graph isomorphism, dominating set, clique, and colorability. We negatively answer the 25-year-old question of Biro, Hujter, and Tuza which asks whether H-graphs can be recognized in polynomial time, for a fixed graph H. We prove that it is NP-complete if H contains the diamond graph as a minor. On the positive side, we provide a polynomial-time algorithm recognizing T-graphs, for each fixed tree T. For the special case when T is a star S-d of degree d, we have an O(n(3.5))-time algorithm. We give FPT- and XP-time algorithms solving the minimum dominating set problem on S-d-graphs and H-graphs, parametrized by d and the size of H, respectively. The algorithm for H-graphs adapts to an T-time algorithm for the independent set and the independent dominating set problems on H-graphs. If H contains the double-triangle as a minor, we prove that the graph isomorphism problem is G(I)-complete and that the clique problem is APX-hard. On the positive side, we show that the clique problem can be solved in polynomial time if H is a cactus graph. Also, when a graph has a Helly H-representation, the clique problem is polynomial-time solvable. Further, we show that both the k-clique and the list k-coloring problems are solvable in FPT-time on H-graphs, parameterized by k and the treewidth of H. In fact, these results apply to classes of graphs with treewidth bounded by a function of the clique number. We observe that H-graphs have at most n(O(parallel to H parallel to)) minimal separators which allows us to apply the meta-algorithmic framework of Fomin, Todinca, and Villanger (2015) to show that for each fixed t, finding a maximum induced sub-graph of treewidth t can be done in polynomial time. In the case when H is a cactus, we improve the bound to O(parallel to H parallel to n(2)).
Original languageEnglish
Pages (from-to)3281-3318
Number of pages38
Issue number11
Early online date23 Jul 2021
Publication statusPublished - Nov 2021


  • H-graphs
  • Recognition
  • NP-completess
  • Graph isomorphism
  • Dominating set
  • Maximum clique
  • Coloring
  • Treewidth
  • Minimal separators


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