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 \(O(\Vert H\Vert n^2)\).

比罗等人。(Discrete. Math 100(1–3):267–279, 1992) 介绍了H图的概念，即图H的细分的连通子图的交集图。它们涉及并概括了许多重要的几何交集图类，例如区间图、圆弧图、分裂图和弦图。我们的论文通过研究H图上的几个经典计算问题，开始了几何相交图领域的新研究：识别、图同构、支​​配集、派系和可着色性。我们负回答比罗，Hujter和土杂的25岁的问题，询问是否^ h -graphs可以在多项式时间内得到认可，对于固定的图^ h. 我们证明它是\(\textsf {NP}\) -如果H包含菱形图作为次要。从积极的方面来说，我们为每个固定的树T提供了一种识别T图的多项式时间算法。对于T是度数为d的恒星\(S_d\)的特殊情况，我们有一个\(\mathcal{O}(n^{3.5})\)时间算法。我们给出了\(\textsf {FPT}\) - 和\(\textsf {XP}\) - 时间算法来解决\(S_d\) -graphs 和H -graphs上的最小支配集问题，由d和大小参数化的H，分别。该算法用于ħ -graphs适应一个\（\ textsf {XP} \） -time算法的独立集和在独立控制集问题ħ -graphs。如果H包含双三角形作为次要，我们证明图同构问题是GI 完全的，并且团问题是APX难的。从积极的方面来说，我们表明如果H是仙人掌图，可以在多项式时间内解决集团问题。此外，当图具有 Helly H表示时，集团问题是多项式时间可解的。此外，我们证明了k-clique 和 list k -coloring 问题可以在H -graphs上的FPT -time中解决，由k和H的树宽参数化。事实上，这些结果适用于树宽以团数函数为界的图类。我们观察到H 图最多有\(n^{O(\Vert H\Vert )}\) 个最小的分隔符，这允许我们应用 Fomin、Todinca 和 Villanger（2015）的元算法框架来证明对于每个固定的t，可以在多项式时间内找到树宽t的最大诱导子图。在H的情况下 是仙人掌，我们改进了\(O(\Vert H\Vert n^2)\)的界限。