Inserting an edge into a planar graph
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
Improved Approximations of Crossings in Graph Drawings and VLSI Layout Areas
SIAM Journal on Computing
Crossing-Critical Graphs and Path-Width
GD '01 Revised Papers from the 9th International Symposium on Graph Drawing
Computing crossing numbers in quadratic time
Journal of Computer and System Sciences - STOC 2001
SIAM Journal on Discrete Mathematics
Crossing number is hard for cubic graphs
Journal of Combinatorial Theory Series B
Inserting a vertex into a planar graph
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Approximating the Crossing Number of Apex Graphs
Graph Drawing
Approximating the crossing number of toroidal graphs
ISAAC'07 Proceedings of the 18th international conference on Algorithms and computation
Adding one edge to planar graphs makes crossing number hard
Proceedings of the twenty-sixth annual symposium on Computational geometry
Approximating the crossing number of graphs embeddable in any orientable surface
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Algorithms and theory of computation handbook
Do we really understand the crossing numbers?
MFCS'10 Proceedings of the 35th international conference on Mathematical foundations of computer science
A tighter insertion-based approximation of the crossing number
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
Facets in the Crossing Number Polytope
SIAM Journal on Discrete Mathematics
Vertex insertion approximates the crossing number of apex graphs
European Journal of Combinatorics
On graph crossing number and edge planarization
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Advances in the planarization method: effective multiple edge insertions
GD'11 Proceedings of the 19th international conference on Graph Drawing
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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.