Given an n-vertex graph G, a drawing of G in the plane is a mapping of its vertices into points of the plane, and its edges into continuous curves, connecting the images of their endpoints. A crossing in such a drawing is a point where two such curves intersect. In the Minimum Crossing Number problem, the goal is to find a drawing of G with minimum number of crossings. The value of the optimal solution, denoted by OPT, is called the graph's crossing number. This is a very basic problem in topological graph theory, that has received a significant amount of attention, but is still poorly understood algorithmically. The best currently known efficient algorithm produces drawings with @math crossings on bounded-degree graphs, while only a constant factor hardness of approximation is known. A closely related problem is Minimum Edge Planarization, in which the goal is to remove a minimum-cardinality subset of edges from G, such that the remaining graph is planar. Our main technical result establishes the following connection between the two problems: if we are given a solution of cost k to the Minimum Edge Planarization problem on graph G, then we can efficiently find a drawing of G with at most @math crossings, where @math is the maximum degree in G. This result implies an @math -approximation for Minimum Crossing Number, as well as improved algorithms for special cases of the problem, such as, for example, k-apex and bounded-genus graphs.
@cite_0: 
@cite_35: We show that for every fixed k ≥ 0 there is a quadratic time algorithm that decides whether a given graph has crossing number at most k and, if this is the case, computes a drawing of the graph into the plane with at most k crossings.
@cite_10: We show that for every fixed k, there is a linear time algorithm that decides whether or not a given graph has crossing number at most k, and if this is the case, computes a drawing of the graph in the plane with at most k crossings. This answers the question posed by Grohe (STOC'01 and JCSS 2004). Our algorithm can be viewed as a generalization of the seminal result by Hopcroft and Tarjan lin1, which determines if a given graph is planar in linear time. Our algorithm can also be compared to the algorithms by Mohar (STOC'96 and Siam J. Discrete Math 2001), for testing the embeddability of an input graph in a fixed surface. For each surface s, Mohar describes an algorithm which yields either an embedding of G in s or a minor of G which is not embeddable in s and is minimal with this property. The same approach allows us to obtain linear time algorithms for the same question for a variety of other crossing numbers. We can also determine in linear time if an input graph can be made planar by the deletion of k edges (for fixed k).
@cite_22: For an arbitrary fixed surface S, a linear time algorithm is presented that for a given graph G either finds an embedding of G in S or identifies a subgraph of G that is homeomorphic to a minimal forbidden subgraph for embeddability in S. A side result of the proof of the algorithm is that minimal forbidden subgraphs for embeddability in S cannot be arbitrarily large. This yields a constructive proof of the result of Robertson and Seymour that for each closed surface there are only finitely many minimal forbidden subgraphs. The results and methods of this paper can be used to solve more general embedding extension problems.
Grohe @cite_35 , answering a question of Downey and Fellows @cite_0 , proved that the crossing number is fixed-parameter tractable. In particular, for any fixed number of crossings his algorithm computes an optimal drawing in @math time. Building upon the breakthrough result of Mohar @cite_22 for embedding graphs into a surface of bounded genus, Kawarabayashi and Reed @cite_10 gave an improved fixed-parameter algorithm with running time @math .