Degenerate Crossing Numbers

Crossing Numbers

For any simple graph G = (V, E), we can define four types of crossing number: crossing number, rectilinear crossing number, odd-crossing number, and pairwise crossing number. We discuss the relationship of them.

Biplanar crossing numbers. II. Comparing crossing numbers and biplanar crossing numbers using the probabilistic method

The biplanar crossing number cr2(G) of a graph G is min{cr(G1)+ cr(G2)}, where cr is the planar crossing number and G1 ∪ G2 = G. We show that cr2(G) ≤ (3/8)cr(G). Using this result recursively, we bound the thickness by Θ(G) − 2 ≤ Kcr2(G) log2 n with some constant K. A partition realizing this bound for the thickness can be obtained by a polynomial time randomized algorithm. We show that for an...

Crossing numbers

Crossing Numbers

The crossing number of a graph G is the minimum number of crossings in a drawing of G. The determination of the crossing number is an NP-complete problem. We present two general lower bounds for the crossing number, and survey their applications and generalizations.

On the Degenerate Crossing Number

The degenerate crossing number cr∗(G) of a graph G is the minimum number of crossing points of edges in any drawing of G as a simple topological graph in the plane. This notion was introduced by Pach and Tóth who showed that for a graph G with n vertices and e ≥ 4n edges cr∗(G) = Ω(e/n). In this paper we completely resolve the main open question about degenerate crossing numbers and show that c...