Adjacency Labelling for Planar Graphs (and Beyond)

We show that there exists an adjacency labelling scheme for planar graphs where each vertex of an <tex>$n$</tex>-vertex planar graph <tex>$G$</tex> is assigned a <tex>$(1+o(1))\text{log}_{2}n$</tex>-bit label and the labels of two vertices <tex>$u$</tex> and <tex>$v$</tex> are sufficient to determine if <tex>$uv$</tex> is an edge of <tex>$G$</tex>. This is optimal up to the lower order term and is the first such asymptotically optimal result. An alternative, but equivalent, interpretation of this result is that, for every positive integer <tex>$n$</tex>, there exists a graph <tex>$U_{n}$</tex> with <tex>$n^{1+o(1)}$</tex> vertices such that every <tex>$n$</tex>-vertex planar graph is an induced subgraph of <tex>$U_{n}$</tex>. These results generalize to a number of other graph classes, including bounded genus graphs, apex-minor-free graphs, bounded-degree graphs from minor closed families, and <tex>$k$</tex>-planar graphs.

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

[2]  Robert E. Tarjan,et al.  Making data structures persistent , 1986, STOC '86.

[3]  Leonidas J. Guibas,et al.  Fractional cascading: I. A data structuring technique , 1986, Algorithmica.

[4]  Stephen Alstrup,et al.  Simpler, faster and shorter labels for distances in graphs , 2015, SODA.

[5]  Pat Morin A Fast Algorithm for the Product Structure of Planar Graphs , 2020, ArXiv.

[6]  Mikkel Thorup,et al.  Adjacency Labeling Schemes and Induced-Universal Graphs , 2014, STOC.

[7]  Pat Morin,et al.  Clustered 3-colouring graphs of bounded degree , 2020, Comb. Probab. Comput..

[8]  Petra Scheffler,et al.  Optimal Embedding of a Tree into an Interval Graph in Linear Time , 1992 .

[9]  Inge Li Gørtz,et al.  Distance Labeling Schemes for Trees , 2015, ICALP.

[10]  Stephen Alstrup,et al.  Improved labeling scheme for ancestor queries , 2002, SODA '02.

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

[12]  Peter Elias,et al.  Universal codeword sets and representations of the integers , 1975, IEEE Trans. Inf. Theory.

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

[14]  Mathias Bæk Tejs Knudsen,et al.  Near-Optimal Induced Universal Graphs for Bounded Degree Graphs , 2017, ICALP.

[15]  John Harold Muller,et al.  Local structure in graph classes , 1988 .

[16]  David Adjiashvili,et al.  Labeling Schemes for Bounded Degree Graphs , 2014, ICALP.

[17]  Fan Chung Graham,et al.  Universal graphs and induced-universal graphs , 1990, J. Graph Theory.

[18]  Mathias Bæk Tejs Knudsen,et al.  Sublinear Distance Labeling , 2015, ESA.

[19]  Noga Alon,et al.  Asymptotically optimal induced universal graphs , 2017 .

[20]  Marthe Bonamy,et al.  Shorter Labeling Schemes for Planar Graphs , 2019, SODA.

[21]  Pat Morin,et al.  The structure of k-planar graphs , 2019, ArXiv.

[22]  Mathias Bæk Tejs Knudsen,et al.  Optimal Induced Universal Graphs and Adjacency Labeling for Trees , 2015, 2015 IEEE 56th Annual Symposium on Foundations of Computer Science.

[23]  C. Nash-Williams Edge-disjoint spanning trees of finite graphs , 1961 .

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

[25]  Morin Pat Open Data Structures , 2013 .