# New Results on Edge Partitions of 1-plane Graphs

## TL;DR

This paper investigates how to partition edges of 1-plane graphs into two plane graphs with degree constraints, proving optimal bounds, NP-completeness, and efficient algorithms for specific cases, with implications for graph drawing.

## Contribution

It establishes degree bounds for NIC-plane graphs, proves NP-completeness for degree two partitions, and provides quadratic-time algorithms for degree one partitions.

## Key findings

- NIC-plane graphs admit degree three edge partitions, which is optimal.
- Deciding degree two edge partitions is NP-complete.
- Degree one edge partitions can be decided and constructed in quadratic time.

## Abstract

A $1$-plane graph is a graph embedded in the plane such that each edge is crossed at most once. A NIC-plane graph is a $1$-plane graph such that any two pairs of crossing edges share at most one end-vertex. An edge partition of a $1$-plane graph $G$ is a coloring of the edges of $G$ with two colors, red and blue, such that both the graph induced by the red edges and the graph induced by the blue edges are plane graphs. We prove the following: $(i)$ Every NIC-plane graph admits an edge partition such that the red graph has maximum vertex degree three; this bound on the vertex degree is worst-case optimal. $(ii)$ Deciding whether a $1$-plane graph admits an edge partition such that the red graph has maximum vertex degree two is NP-complete. $(iii)$ Deciding whether a $1$-plane graph admits an edge partition such that the red graph has maximum vertex degree one, and computing one in the positive case, can be done in quadratic time. Applications of these results to graph drawing are also discussed.