Exact Crossing Number Parameterized by Vertex Cover

Abstract

We prove that the exact crossing number of a graph can be efficiently computed for simple graphs having bounded vertex cover. In more precise words, Crossing Number is in FPT when parameterized by the vertex cover size. This is a notable advance since we know only very few nontrivial examples of graph classes with unbounded and yet efficiently computable crossing number. Our result can be viewed as a strengthening of a previous result of Lokshtanov [arXiv, 2015] that Optimal Linear Arrangement is in FPT when parameterized by the vertex cover size, and we use a similar approach of reducing the problem to a tractable instance of Integer Quadratic Programming as in Lokshtanov's paper.

Grid Recognition: Classical and Parameterized Computational Perspectives

“…Research at the intersection of graph drawing and parameterized complexity (and parameterized algorithms in particular) is in its infancy. Most (in particular, the early efforts) have been directed at variants of the classic Crossing Minimization problem, introduced by Turán in 1940 [56], parameterized by the number of crossings (see, e.g., [34,46,26,39,40,47]). However, in the past few years, there is an increasing interest in the analysis of a variety of other problems in graph drawing from the perspective of parameterized complexity (see, e.g., [10,2,35,14,38,6,21,20,11,23,49,48] and the upcoming Dagstuhl seminar [1]).…”