Clustered planarity testing revisited
Summary: 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. \textit{G. Di Battista} and \textit{F. Frati} [J. Graph Algorithms Appl. 13, No. 3, 349--378 (2009; Zbl 1184.68355)] 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.
scientific article; zbMATH DE number 6513825
scientific article