On the one-sided crossing minimization in a bipartite graph with large degrees

Given a bipartite graph G=(V,W,E), a 2-layered drawing consists of placing nodes in the first node set V on a straight line L1 and placing nodes in the second node set W on a parallel line L2. For a given ordering of nodes in W on L2, the one-sided crossing minimization problem asks to find an ordering of nodes in V on L1 so that the number of arc crossings is minimized. A well-known lower bound LB on the minimum number of crossings is obtained by summing up min{cuv,cvu} over all node pairs u,v∈V, where cuv denotes the number of crossings generated by arcs incident to u and v when u precedes v in an ordering. In this paper, we prove that there always exists a solution whose crossing number is at most (1.2964+12/(δ-4))LB if the minimum degree δ of a node in V is at least 5.