The structure of k-planar graphs

Dujmovic et al. (FOCS 2019) recently proved that every planar graph is a subgraph of the strong product of a graph of bounded treewidth and a path. This tool has been used to solve longstanding problems on queue layouts, non-repetitive colouring, $p$-centered colouring, and implicit graph encoding. We generalise this result for $k$-planar graphs, where a graph is $k$-planar if it has a drawing in the plane in which each edge is involved in at most $k$ crossings. In particular, we prove that every $k$-planar graph is a subgraph of the strong product of a graph of treewidth $O(k^5)$ and a path. This is the first result of this type for a non-minor-closed class of graphs. It implies, amongst other results, that $k$-planar graphs have non-repetitive chromatic number upper-bounded by a function of $k$. All these results generalise for drawings of graphs on arbitrary surfaces. In fact, we work in a much more general setting based on so-called shortcut systems that are of independent interest.

[1]  Michael A. Bekos,et al.  Queue Layouts of Planar 3-Trees , 2018, Algorithmica.

[2]  János Pach,et al.  Recognizing String Graphs Is Decidable , 2001, GD.

[3]  David R. Wood,et al.  Nonrepetitive colouring via entropy compression , 2011, Comb..

[4]  David R. Wood,et al.  Planar graphs have bounded nonrepetitive chromatic number , 2019, ArXiv.

[5]  Pat Morin,et al.  Planar Graphs have Bounded Queue-Number , 2019, 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS).

[6]  David R. Wood,et al.  Improper colourings inspired by Hadwiger's conjecture , 2017, J. Lond. Math. Soc..

[7]  Giuseppe Liotta,et al.  An annotated bibliography on 1-planarity , 2017, Comput. Sci. Rev..

[8]  Prosenjit Bose,et al.  PROXIMITY GRAPHS: E, δ, Δ, χ AND ω , 2012, Int. J. Comput. Geom. Appl..

[9]  János Pach,et al.  A Separator Theorem for String Graphs and its Applications , 2009, Combinatorics, Probability and Computing.

[10]  Noga Alon,et al.  Nonrepetitive colorings of graphs , 2002, Random Struct. Algorithms.

[11]  Xuding Zhu,et al.  Colouring graphs with bounded generalized colouring number , 2009, Discret. Math..

[12]  Cyril Gavoille,et al.  Shorter Implicit Representation for Planar Graphs and Bounded Treewidth Graphs , 2007, ESA.

[13]  Stefan Felsner,et al.  Improved bounds for centered colorings , 2019, SODA.

[14]  Veit Wiechert,et al.  On the Queue-Number of Graphs with Bounded Tree-Width , 2016, Electron. J. Comb..

[15]  Carsten Thomassen,et al.  Graphs on Surfaces , 2001, Johns Hopkins series in the mathematical sciences.

[16]  Alexander Grigoriev,et al.  Algorithms for Graphs Embeddable with Few Crossings Per Edge , 2005, FCT.

[17]  Fedor V. Fomin,et al.  Bidimensionality and geometric graphs , 2011, SODA.

[18]  János Pach,et al.  Applications of a New Separator Theorem for String Graphs , 2013, Combinatorics, Probability and Computing.

[19]  Michal Pilipczuk,et al.  Polynomial bounds for centered colorings on proper minor-closed graph classes , 2018, SODA.

[20]  Moni Naor,et al.  Implicit representation of graphs , 1992, STOC '88.

[21]  Hans L. Bodlaender,et al.  A Partial k-Arboretum of Graphs with Bounded Treewidth , 1998, Theor. Comput. Sci..

[22]  David Eppstein,et al.  Structure of Graphs with Locally Restricted Crossings , 2015, SIAM J. Discret. Math..

[23]  David R. Wood,et al.  Parameters Tied to Treewidth , 2013, J. Graph Theory.

[24]  Ferran Hurtado,et al.  On crossing numbers of geometric proximity graphs , 2011, Comput. Geom..

[25]  Zhi-Zhong Chen,et al.  Map graphs , 1999, JACM.

[26]  B. A. Reed,et al.  Algorithmic Aspects of Tree Width , 2003 .

[27]  F. Thomas Leighton,et al.  Complexity Issues in VLSI , 1983 .

[28]  Pat Morin,et al.  Layered separators in minor-closed graph classes with applications , 2013, J. Comb. Theory, Ser. B.

[29]  Michael A. Bekos,et al.  Queue Layouts of Planar 3-Trees , 2020, Algorithmica.

[30]  Arnold L. Rosenberg,et al.  Comparing Queues and Stacks as Mechanisms for Laying out Graphs , 1992, SIAM J. Discret. Math..

[31]  Ivan Stojmenovic,et al.  Routing with Guaranteed Delivery in Ad Hoc Wireless Networks , 1999, DIALM '99.