IC-planar graphs are those graphs that admit a drawing where no two crossed edges share an end-vertex and each edge is crossed at most once. They are a proper subfamily of the 1-planar graphs. Given an embedded IC-planar graph G with n vertices, we present an O(n)-time algorithm that computes a straight-line drawing of G in quadratic area, and an O(n3)-time algorithm that computes a straight-line drawing of G with right-angle crossings in exponential area. Both these area requirements are worst-case optimal. We also show that it is NP-complete to test IC-planarity both in the general case and in the case in which a rotation system is fixed for the input graph. Furthermore, we describe a polynomial-time algorithm to test whether a set of matching edges can be added to a triangulated planar graph such that the resulting graph is IC-planar.
@article{arxiv.1509.00388,
title = {Recognizing and Drawing IC-planar Graphs},
author = {Franz J. Brandenburg and Walter Didimo and William S. Evans and Philipp Kindermann and Giuseppe Liotta and Fabrizio Montecchiani},
journal= {arXiv preprint arXiv:1509.00388},
year = {2016}
}