We study the algorithmic aspect of edge bundling. A bundled crossing in a drawing of a graph is a group of crossings between two sets of parallel edges. The bundled crossing number is the minimum number of bundled crossings that group all crossings in a drawing of the graph. We show that the bundled crossing number is closely related to the orientable genus of the graph. If multiple crossings a...

The minor crossing number of a graph G is defined as the minimum crossing number of all graphs that contain G as a minor. Basic properties of this new invariant are presented. We study topological structure of graphs with bounded minor crossing number and obtain a new strong version of a lower bound based on the genus. We also give a generalization of an inequality of Moreno and Salazar crossin...

Improved bounds for the crossing number of the mesh of trees graph, Mn, are derived. In particular, we derive a new lower bound of 5n log n−44n 80 ‡ which improves on the previous bound of Leighton [11] by a constant factor, and an upper bound of (log n− 10 3 )n 2 + 8n− 20 3 . In addition, we construct drawings of Mn which achieve the upper bound number of crossings. We also prove that the cros...

The best lower bound known on the crossing number of the complete bipartite graph is : cr(Km,n) ≥ (1/5)(m)(m − 1)bn/2cb(n − 1)/2c In this paper we prove that: cr(Km,n) ≥ (1/5)m(m − 1)bn/2cb(n − 1)/2c + 9.9 × 10−6m2n2 for sufficiently large m and n.

The bipartite crossing number of a bipartite graph is the minimum number of crossings of edges when the partitions are placed on two parallel lines and edges are drawn as straigh line segments between the lines. We prove exact results, asymtotics and new upper bounds for the bipartite crossing numbers of 2-dimensional mesh graphs. Especially we show that bcr(P6 × Pn) = 35n − 47, for n ≥ 7.

We characterize Pfaffian graphs in terms of their drawings in the plane. We generalize the techniques used in the proof of this characterization, and prove a theorem about the numbers of crossings in T -joins in different drawings of a fixed graph. As a corollary we give a new proof of a theorem of Kleitman on the parity of crossings in drawings of K2j+1 and K2j+1,2k+1.

The minor crossing number of a graph G is the minimum crossing number of a graph that contains G as a minor. It is proved that for every graph H there is a constant c, such that every graph G with no H-minor has minor crossing number at most c|V (G)|.