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

Abstract

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 L 1 and placing nodes in the second node set W on a parallel line L 2 . For a given ordering of nodes in W on L 2 , the one-sided crossing minimization problem asks to find an ordering of nodes in V on L 1 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 { c uv , c vu } over all node pairs u , v ∈ V , where c uv 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.