LIPIcs.GD.2024.37.pdf
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The rectilinear crossing number of G is the minimum number of crossings in a straight-line drawing of G. A single-crossing graph is a graph whose crossing number is at most one. We prove that every n-vertex graph G that excludes a single-crossing graph as a minor has rectilinear crossing number O(Δ n), where Δ is the maximum degree of G. This dependence on n and Δ is best possible. The result applies, for example, to K₅-minor-free graphs, and bounded treewidth graphs. Prior to our work, the only bounded degree minor-closed families known to have linear rectilinear crossing number were bounded degree graphs of bounded treewidth as well as bounded degree K_{3,3}-minor-free graphs. In the case of bounded treewidth graphs, our O(Δ n) result is again tight and it improves on the previous best known bound of O(Δ² n) by Wood and Telle, 2007.
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