A Unified FPT Framework for Crossing Number Problems

Abstract: The basic (and traditional) crossing number problem is to determine the minimum number of crossings in a topological drawing of an input graph in the plane. In this paper, we develop a unified framework that yields fixed-parameter tractable (FPT) algorithms for many generalized crossing number problems in the plane and on surfaces. Our main result takes the following form. We fix a surface S, an integer r, and a map \kappa from the set of topological drawings of graphs in S to the nonnegative integers, essentially describing which types of drawings we accept and how we want to count the crossings in them. Then deciding whether a graph G has a drawing D on S with \kappa(D)<=r can be done in time quadratic in the size of G. More generally, we may take as input an edge-colored graph, and distinguish crossings by the colors of the involved edges; and we may allow to perform a bounded number of edge removals and vertex splits to G before drawing it. The proof is a reduction to the embeddability of a graph on a two-dimensional simplicial complex, which admits an FPT algorithm by a result of Colin de Verdi`ere and Magnard [ESA 2021]. As a direct consequence, we obtain, in a unified way, FPT algorithms for many established topological crossing number variants, including, e.g., fixed-rotation and color-constrained ones. While some of these variants already had previously published FPT algorithms, for many of them we obtain algorithms with a better runtime. Moreover, our framework applies to drawings in any fixed surface and, for instance, solves the natural surface generalizations of k-planar, k-quasi-planar, min-k-planar, and k-gap crossing numbers, as well as the joint and the edge crossing numbers, the skewness, and the splitting number in surfaces, all of which except the splitting number have not been handled in surfaces other than the plane yet.