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. We develop a unified framework yielding fixed-parameter tractable (FPT) algorithms for many generalized crossing number problems. Our framework takes the following form. We fix a surface S and a class D of "allowed" topological drawings of graphs in S (e.g., some class of drawings with at most t crossings). We assume that testing membership in D can be done algorithmically, and that restricting a drawing in D, extending it without adding any crossing, or transforming it with a self-homeomorphism of S yields a drawing that is also in D. Then deciding whether an input graph G has a drawing in D, and computing one if it is the case, is fixed-parameter tractable in (essentially) the genus of S and the maximum number of crossings in a drawing in D. 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 a bounded number of edge removals and vertex splits on G before drawing it. The proof is a reduction to the embeddability of a graph on a two-dimensional simplicial complex. This implies, in a unified way, linear or quadratic FPT algorithms for many topological crossing number variants established in the graph drawing community. Some of these variants already had previously published FPT algorithms, mostly relying on Courcelle's metatheorem, but our algorithms have a better runtime. Moreover, our framework extends to these crossing number variants in any fixed surface, and also allows us to fix the rotation system of the drawing of a graph in some variants.