Abstract
The weak variant of the Hanani–Tutte theorem says that a graph is planar, if it can be drawn in the plane so that every pair of edges cross an even number of times. Moreover, we can turn such a drawing into an embedding without changing the order in which edges leave the vertices. We prove a generalization of the weak Hanani–Tutte theorem that also easily implies the monotone variant of the weak Hanani–Tutte theorem by Pach and Tóth. Thus, our result can be thought of as a common generalization of these two neat results. In other words, we prove the weak Hanani-Tutte theorem for strip clustered graphs, whose clusters are linearly ordered vertical strips in the plane and edges join only vertices in the same cluster or in neighboring clusters with respect to this order.
Besides usual tools for proving Hanani-Tutte type results our proof combines Hall’s marriage theorem, and a characterization of embedded upward planar digraphs due to Bertolazzi et al.
The author gratefully acknowledges support from the Swiss National Science Foundation Grant No. 200021-125287/1 GIG/11/E023.
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Notes
- 1.
This type of clustered graphs is usually called flat clustered graph in the graph drawing literature. We chose this simplified notation in order not to overburden the reader with unnecessary notation.
- 2.
The argument in the proof of Theorem 3 proves, in fact, a strong variant even in the case, when we require the vertices participating in a cut or two-cut to have the maximum degree three. Hence, we obtained a polynomial time algorithm even in the case of sub-cubic cuts and two-cuts.
- 3.
We would not have to do anything that follows, if we could turn all the faces into simple ones. However, this seems to be a difficult task.
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Acknowledgment
We would like to express our special thanks of gratitude to the organizers and participants of the 11th GWOP workshop, where we could discuss the research problems treated in the present paper. In particular, we especially benefited from the discussions with Bettina Speckmann, Edgardo Roldán-Pensado and Sebastian Stich. Furthermore, we would like to thank Ján Kynčl for useful discussions at the initial stage, and Gábor Tardos for comments at the final stage of this work.
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Fulek, R. (2014). Towards the Hanani-Tutte Theorem for Clustered Graphs. In: Kratsch, D., Todinca, I. (eds) Graph-Theoretic Concepts in Computer Science. WG 2014. Lecture Notes in Computer Science, vol 8747. Springer, Cham. https://doi.org/10.1007/978-3-319-12340-0_15
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