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On H-Topological Intersection Graphs

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Biró 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 Biró, Hujter, and Tuza which asks whether H-graphs can be recognized in polynomial time, for a fixed graph H. We prove that it is \(\textsf {NP}\)-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 \(\mathcal{O}(n^{3.5})\)-time algorithm. We give \(\textsf {FPT}\)- and \(\textsf {XP}\)-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 \(\textsf {XP}\)-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 GI-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 \(n^{O(\Vert H\Vert )}\) 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 subgraph of treewidth t can be done in polynomial time. In the case when H is a cactus, we improve the bound to \(O(\Vert H\Vert n^2)\).

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Notes

  1. where each node of T corresponds to a maximal clique of G

  2. The diamond graph is obtained by deleting an edge from a 4-vertex clique.

  3. While the representation of a vertex of \(G_{\mathcal{P}}\) might “reach out” beyond \(D^*\) onto an incident edge, it can never traverse all of such an edge because, by Lemma 3.1, there is a vertex \(x_e\) of \(H_3\) occupying the “middle” of that edge and, by construction, \(x_e\) is not adjacent to any vertex of \(G_{\mathcal{P}}\).

  4. Note that this is just a representational convenience for dynamic programming.

  5. In our prior work [17], we referred to this as being treewidth-bounded, but have changed the name to be consistent with other parameter-treewidth bounds given in bidimensionality theory [18].

  6. i.e., minimum feedback vertex set

  7. Informally, CMSO consists of all logic formulas with quantifiers over vertices, edges, edge sets and vertex sets, and counting modulo constants. For more information on this logic see, e.g., [11]. Note: in [11], this logic is abbreviated by CMS\(_2\) instead of CMSO as in [23].

  8. A similar result with a slightly better bound is given in a recent manuscript, see [22]. Our proof and theirs seem to follow similar reasoning, but have been obtained independently, as also noted in [22].

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Acknowledgements

We would like to thank Pavel Klavík for suggesting to study of H-graphs and for several helpfull discussions. We would also like to thank the DIMACS REU 2015 program, held at the Rutgers University, where the whole project started.

Funding

P. Zeman: Supported by GAUK 1224120 and by GAČR 19-17314J.

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Correspondence to Peter Zeman.

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This paper is the combination and extension of the conference versions which appeared at WG 2017 [16] and Eurocomb 2017 [17].

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Chaplick, S., Töpfer, M., Voborník, J. et al. On H-Topological Intersection Graphs. Algorithmica 83, 3281–3318 (2021). https://doi.org/10.1007/s00453-021-00846-3

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