Crossing number of graphs with rotation systems

Abstract

We show that computing the crossing number of a graph with a given rotation system is NP-complete. This result leads to a new and much simpler proof of Hliněný's result, that computing the crossing number of a cubic graph (without rotation system) is NP-complete. We also investigate the special case of multigraphs with rotation systems on a fixed number k of vertices. For k∈=∈1 and k∈=∈2 the crossing number can be computed in polynomial time and approximated to within a factor of 2 in linear time. For larger k we show how to approximate the crossing number to within a factor of in time O(m k∈+∈2) on a graph with m edges. © 2008 Springer-Verlag Berlin Heidelberg.