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arXiv · 1701.04375

NIC-Planar Graphs

Christian Bachmaier·Franz J. Brandenburg·Kathrin Hanauer·Daniel Neuwirth·Josef Reislhuber
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Abstract

A graph is NIC-planar if it admits a drawing in the plane with at most one crossing per edge and such that two pairs of crossing edges share at most one common end vertex. NIC-planarity generalizes IC-planarity, which allows a vertex to be incident to at most one crossing edge, and specializes 1-planarity, which only requires at most one crossing per edge. We characterize embeddings of maximal NIC-planar graphs in terms of generalized planar dual graphs. The characterization is used to derive tight bounds on the density of maximal NIC-planar graphs which ranges between 3.2(n-2) and 3.6(n-2). Further, we prove that optimal NIC-planar graphs with 3.6(n-2) edges have a unique embedding and can be recognized in linear time, whereas the general recognition problem of NIC-planar graphs is NP-complete. In addition, we show that there are NIC-planar graphs that do not admit right angle crossing drawings, which distinguishes NIC-planar from IC-planar graphs.

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Christian Bachmaier, Franz J. Brandenburg, Kathrin Hanauer, Daniel Neuwirth, Josef Reislhuber. 2017-01-16. NIC-Planar Graphs. https://arxiv.org/abs/1701.04375

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