Bundled crossings in embedded graphs

Edge crossings in a graph drawing are an important factor in the drawing’s quality. However, it is not just the presence of crossings that determines the drawing’s quality: any drawing of a nonplanar graph in the plane necessarily contains crossings, but the geometric structure of those crossings can have a significant impact on the drawing’s readability. In particular, the structure of two disjoint groups of locally parallel edges (bundles) intersecting in a complete crossbar (a bundled crossing) is visually simpler—even if it involves many individual crossings—than an equal number of random crossings scattered in the plane. In this paper, we investigate the complexity of partitioning the crossings of a given drawing of a graph into a minimum number of bundled crossings. We show that this problem is NP-hard, propose a constant-factor approximation scheme for the case of circular embeddings, where all vertices lie on the outer face, and show that the bundled crossings problem in general graphs is related to a minimum dissection problem.