Bounds for rectilinear crossing numbers
Journal of Graph Theory
The crossing number of K3,n in a surface
Journal of Graph Theory
Graph minors. XVI. excluding a non-planar graph
Journal of Combinatorial Theory Series B
SIAM Journal on Discrete Mathematics
Discrete Applied Mathematics - Structural decompositions, width parameters, and graph labelings (DAS 5)
Crossing number of toroidal graphs
GD'05 Proceedings of the 13th international conference on Graph Drawing
Planar crossing numbers of genus g graphs
ICALP'06 Proceedings of the 33rd international conference on Automata, Languages and Programming - Volume Part I
Approximating the crossing number of toroidal graphs
ISAAC'07 Proceedings of the 18th international conference on Algorithms and computation
On graph crossing number and edge planarization
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
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Tree decompositions of graphs are of fundamental importance in structural and algorithmic graph theory. Planar decompositions generalise tree decompositions by allowing an arbitrary planar graph to index the decomposition. We prove that every graph that excludes a fixed graph as a minor has a planar decomposition with bounded width and a linear number of bags. The crossing number of a graph is the minimum number of crossings in a drawing of the graph in the plane.We prove that planar decompositions are intimately related to the crossing number, in the sense that a graph with bounded degree has linear crossing number if and only if it has a planar decomposition with bounded width and linear order. It follows from the above result about planar decompositions that every graph with bounded degree and an excluded minor has linear crossing number. Analogous results are proved for the convex and rectilinear crossing numbers. In particular, every graph with bounded degree and bounded tree-width has linear convex crossing number, and every K3, 3-minor-free graph with bounded degree has linear rectilinear crossing number.