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.

Highlights

  • Investigation of graph planarity can be traced back to the 1930s and developments accomplished at that time by Hanani [22], Kuratowski [27], Whitney [39] and others

  • Di Battista and Frati proved that clustered planarity of embedded clustered graphs whose every face is incident to at most five vertices can be tested in polynomial time

  • The Hanani–Tutte theorem says that a graph is planar if it can be drawn in the plane so that no pair of independent edges crosses an odd number of times

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Summary

Introduction

Investigation of graph planarity can be traced back to the 1930s and developments accomplished at that time by Hanani [22], Kuratowski [27], Whitney [39] and others. The (strong) Hanani–Tutte theorem says that a graph is planar if it can be drawn in the plane so that no pair of independent edges crosses an odd number of times. Its variant known as the weak Hanani–Tutte theorem [4, 31, 34] states that if G has a drawing D where every pair of edges cross an even number of times, G has an embedding that preserves the cyclic order of edges at vertices in D. We prove a variant of the (strong) Hanani–Tutte theorem for flat clustered graphs with two clusters forming a partition of the vertex set. We give an alternative polynomial-time algorithm for deciding c-planarity of embedded flat clustered graphs with small faces, reproving a result of Di Battista and Frati [10].

Algorithm
Weak Hanani–Tutte for two-clustered graphs
Proof of Theorem 1
Strong Hanani–Tutte for two-clustered graphs
Proof of Theorem 2
Strong Hanani–Tutte for c-connected clustered graphs
Counterexample on three clusters
Proof of Theorem 4
Small faces
Proof of Theorem 5
Concluding remarks
Full Text
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