In a total order of the vertices of a graph, two edges with no endpoint in common can be crossing, nested, or disjoint. A k-stack (resp...

The geometric thickness of a graph is the minimum number of layers such that the graph can be drawn in the plane with edges as straight-line segments, and with each edge assigned to a layer so that no two edges on the same layer cross. We consider a variation on this theme in which each edge is allowed one bend. We prove that the vertices of an n-vertex m-edge graph can be positioned in a $$ grid and the edges assigned to $$ layers, so that each edge is drawn with at most one bend and no two edges on the same layer cross. The proof is a 2-dimensional generalization of a theorem of S. M. Malitz [J. Algorithms 17(1):71-84, 1994] on book embeddings. We obtain a Las Vegas algorithm to compute the drawing in O(m log n log log n) time (with high probability).