# NIC-Planar Graphs

## TL;DR

This paper introduces NIC-planar graphs, characterizes their embeddings, establishes density bounds, and explores recognition complexity and geometric properties, advancing understanding of this graph class.

## Contribution

It provides a characterization of maximal NIC-planar graphs, tight density bounds, and complexity results for recognition and embedding uniqueness.

## Key findings

- Maximal NIC-planar graphs have density between 3.2(n-2) and 3.6(n-2).
- Recognition of optimal NIC-planar graphs is linear-time solvable.
- NIC-planar graphs can lack right angle crossing drawings, unlike IC-planar graphs.

## 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.