On the crossing number of almost planar graphs

Crossing minimization is one of the most challenging algorithmic problems in topological graph theory, with strong ties to graph drawing applications. Despite a long history of intensive research, no practical "good" algorithm for crossing minimization is known (that is hardly surprising, since the problem itself is NP-complete). Even more surprising is how little we know about a seemingly simple particular problem: to minimize the number of crossings in an almost planar graph, that is, a graph with an edge whose removal leaves a planar graph. This problem is in turn a building block in an "edge insertion" heuristic for crossing minimization. In this paper we prove a constant factor approximation algorithm for the crossing number of almost planar graphs with bounded degree. On the other hand, we demonstrate nontriviality of the crossing minimization problem on almost planar graphs by exhibiting several examples, among them new families of crossing critical graphs which are almost planar and projective.