Clustered planarity testing revisited

TL;DR

This paper extends the Hanani--Tutte theorem to certain classes of clustered graphs, providing new theoretical insights and a simplified proof for a known polynomial-time clustered planarity testing result.

Contribution

It generalizes the Hanani--Tutte theorem to two-clustered graphs, explores limitations for more clusters, and offers a concise proof for polynomial-time planarity testing in specific cases.

Findings

Extended Hanani--Tutte theorem to two-clustered graphs

Identified limitations for three or more clusters

Provided a simplified proof for polynomial-time planarity testing

Abstract

The Hanani--Tutte theorem is a classical result proved for the first time in the 1930s that characterizes planar graphs as graphs that admit a drawing in the plane in which every pair of edges not sharing a vertex cross an even number of times. We generalize this result to clustered graphs with two disjoint clusters, and show that a straightforward extension to flat clustered graphs with three or more disjoint clusters is not possible. For general clustered graphs we show a variant of the Hanani--Tutte theorem in the case when each cluster induces a connected subgraph. Di Battista and Frati proved that clustered planarity of embedded clustered graphs whose every face is incident with at most five vertices can be tested in polynomial time. We give a new and short proof of this result, using the matroid intersection algorithm.