Subexponential algorithms in geometric graphs via the subquadratic grid minor property: the role of local radius

In this paper we investigate the existence of subexponential parameterized algorithms of three fundamental cycle-hitting problems in geometric graph classes. The considered problems, Triangle Hitting (TH), Feedback Vertex Set (FVS), and 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(√(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 direction we provide: - a 2^O(k^3/4·log k)n^O(1)-time algorithm for FVS in contact segment graphs; - a 2^O(√(d)· t^2 log t· k^2/3log k) n^O(1)-time algorithm for TH in K_t,t-free d-DIR graphs; and - a 2^O(k^7/9log^3/2k) n^O(1)-time algorithm for TH in contact segment graphs.