The complexity of computing the cylindrical and the $t$-circle crossing number of a graph

Abstract

A plane drawing of a graph is cylindrical if there exist two concentric circles that contain all the vertices of the graph, and no edge intersects (other than at its endpoints) any of these circles. The cylindrical crossing number of a graph G is the minimum number of crossings in a cylindrical drawing of G. In his influential survey on the variants of the definition of the crossing number of a graph, Schaefer lists the complexity of computing the cylindrical crossing number of a graph as an open question. In this paper we settle this by showing that this problem is NPcomplete. Moreover, we show an analogous result for the natural generalization of the cylindrical crossing number, namely the t-circle crossing number.

Bounding the tripartite-circle crossing number of complete tripartite graphs

“…In the study of crossing-minimal drawings of complete and complete bipartite graphs [11,19], drawings where the vertices are placed on one or two circles, and edges do not cross the circles, are conjectured to be crossing-minimal and are thus of special interest. Such drawings are known as 2-page book embeddings [5,12] and cylindrical drawings [4,7,13,23], respectively. (For more details, refer to Section 5.…”