Clustered Planarity Testing Revisited

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.

Synchronized Planarity with Applications to Constrained Planarity Problems

“…The complexity of the general problem that allows disconnected clusters has been open for 30 years. In that time, many special cases have been shown to be polynomial-time solvable [3,11,14,16] before Fulek and Tóth [15] recently settled Clustered Planarity in P. The core ingredient for this is their O(n 8 ) algorithm for the Atomic Embeddability problem. It has two graphs G and H as input.…”

Synchronized Planarity with Applications to Constrained Planarity Problems

“…The complexity of the general problem that allows disconnected clusters has been open for 30 years. In that time, many special cases have been shown to be polynomial-time solvable [3,12,17,19] before Fulek and Tóth [18] recently settled Clustered Planarity in P. The core ingredient for this is their O(n 8 ) algorithm for the Atomic Embeddability problem. It has two graphs G and H as input.…”