Nonplanarity of Iterated Line Graphs

Abstract

The 1-crossing index of a graph $G$ is the smallest integer $k$ such that the $k$ th iterated line graph of $G$ has crossing number greater than 1. In this paper, we show that the 1-crossing index of a graph is either infinite or it is at most 5. Moreover, we give a full characterization of all graphs with respect to their 1-crossing index.