In this paper we investigate the existence of subexponential parameterized algorithms of three fundamental cycle-hitting problems in geometric graph classes. The considered problems, \textsc{Triangle Hitting} (TH), \textsc{Feedback Vertex Set} (FVS), and \textsc{Odd Cycle Transversal} (OCT) ask for the existence in a graph $G$ of a set $X$ of at most $k$ vertices such that $G-X$ is, respectively, triangle-free, acyclic, or bipartite. Such subexponential parameterized algorithms are known to exist in planar and even $H$-minor free graphs from bidimensionality theory [Demaine et al., JACM 2005], and there is a recent line of work lifting these results to geometric graph classes consisting of intersection of "fat" objects ([Grigoriev et al., FOCS 2022] and [Lokshtanov et al., SODA 2022]). In this paper we focus on "thin" objects by considering intersection graphs of segments in the plane with $d$ possible slopes ($d$-DIR graphs) and contact graphs of segments in the plane. Assuming the ETH, we rule out the existence of algorithms: - solving TH in time $2^{o(n)}$ in 2-DIR graphs; and - solving TH, FVS, and OCT in time $2^{o(\sqrt{n})}$ in $K_{2,2}$-free contact 2-DIR graphs. These results indicate that additional restrictions are necessary in order to obtain subexponential parameterized algorithms for %these problems. In this direction we provide: - a $2^{O(k^{3/4}\cdot \log k)}n^{O(1)}$-time algorithm for FVS in contact segment graphs; - a $2^{O(\sqrt d\cdot t^2 \log t\cdot k^{2/3}\log k)} n^{O(1)}$-time algorithm for TH in $K_{t,t}$-free $d$-DIR graphs; and - a $2^{O(k^{7/9}\log^{3/2}k)} n^{O(1)}$-time algorithm for TH in contact segment graphs.

翻译：本文研究了三类基本环击中问题在几何图类中是否存在次指数参数化算法。所考虑的问题包括：三角形击中（TH）、反馈顶点集（FVS）和奇环横贯（OCT），它们要求在图$G$中找出一个大小至多为$k$的顶点集$X$，使得$G-X$分别成为无三角图、无环图或二部图。根据二维性理论[Demaine等，JACM 2005]，此类次指数参数化算法已知存在于平面图乃至$H$-无子式图中，近期研究进一步将这些结论推广至由"胖"物体交集构成的几何图类[Grigoriev等，FOCS 2022；Lokshtanov等，SODA 2022]。本文聚焦"薄"物体，考虑平面中具有$d$种可能斜率的线段交集图（$d$-DIR图）以及平面中线段的接触图。基于ETH假设，我们排除了以下算法的存在性：- 在2-DIR图中以$2^{o(n)}$时间求解TH；- 在$K_{2,2}$-无接触2-DIR图中以$2^{o(\sqrt{n})}$时间求解TH、FVS和OCT。这些结果表明，为获得上述问题的次指数参数化算法需要附加限制条件。在此方向上，我们提供了：- 用于接触线段图中FVS的$2^{O(k^{3/4}\cdot \log k)}n^{O(1)}$时间算法；- 用于$K_{t,t}$-无$d$-DIR图中TH的$2^{O(\sqrt d\cdot t^2 \log t\cdot k^{2/3}\log k)} n^{O(1)}$时间算法；- 用于接触线段图中TH的$2^{O(k^{7/9}\log^{3/2}k)} n^{O(1)}$时间算法。