Edge Partitions of Optimal $2$-plane and $3$-plane Graphs

TL;DR

This paper investigates how to partition edges of optimal 2-plane and 3-plane graphs into simpler subgraphs, providing algorithms and limitations for such partitions to understand their structure better.

Contribution

It introduces new algorithms and impossibility results for partitioning optimal 2-plane and 3-plane graphs into simpler components, advancing understanding of their structure.

Findings

Partition of a simple optimal 2-plane graph into a 1-plane graph and a forest is impossible.

A partition into a 1-plane graph and two plane forests always exists and is computable in linear time.

Optimal 3-plane graphs with biconnected crossing-free edges can be decomposed into a 2-plane graph and two plane forests in linear time.

Abstract

A topological graph is a graph drawn in the plane. A topological graph is $k$-plane, $k>0$, if each edge is crossed at most $k$ times. We study the problem of partitioning the edges of a $k$-plane graph such that each partite set forms a graph with a simpler structure. While this problem has been studied for $k=1$, we focus on optimal $2$-plane and $3$-plane graphs, which are $2$-plane and $3$-plane graphs with maximum density. We prove the following results. (i) It is not possible to partition the edges of a simple optimal $2$-plane graph into a $1$-plane graph and a forest, while (ii) an edge partition formed by a $1$-plane graph and two plane forests always exists and can be computed in linear time. (iii) We describe efficient algorithms to partition the edges of a simple optimal $2$-plane graph into a $1$-plane graph and a plane graph with maximum vertex degree $12$, or with maximum vertex degree $8$ if the optimal $2$-plane graph is such that its crossing-free edges form a graph with no separating triangles. (iv) We exhibit an infinite family of simple optimal $2$-plane graphs such that in any edge partition composed of a $1$-plane graph and a plane graph, the plane graph has maximum vertex degree at least $6$ and the $1$-plane graph has maximum vertex degree at least $12$. (v) We show that every optimal $3$-plane graph whose crossing-free edges form a biconnected graph can be decomposed, in linear time, into a $2$-plane graph and two plane forests.