On H-Topological Intersection Graphs

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