1. LaBRI, Université Bordeaux I
STACS’03
An Information-Theoretic Upper Bound of Planar Graphs Using Triangulation
I will be talking to you about An Information-Theoretic Upper Bound of Planar Graphs Using Triangulation. This work has been completed by myself, Bertrand Le Saëc and Mohamed Mosbah.
Nicolas Bonichon, Cyril Gavoille & Nicolas Hanusse

2. Planar Graphs (unlabeled)
Maximal planar graph
(or triangulation)
Plane embedding (or plane graph)
First let me define the graphs we will consider in this talk.
A planar graph is a graph with can be embedded in the plane.
Here is an exemple of a planar graph.
A planar graph G is called maximal if no edges can be added to G without breaking the planarity property. Here is an exemple of maximal plane graph.
A plane embedding or plane graph for short, is a graph where the incident edges of each node are ordered. This two plane graphs are two different plane embeddings of the same planar graph.

3. p(n) = number of n-node planar graphs
Our Problem
How many bits are needed to encode a planar graph with n nodes?
What is the number of edges of a uniformly random planar graph?
p(n) = number of n-node planar graphs
p(n)?
The questions that we would like to answer are the following ones:
How many planar graphs with n nodesexists ?
How many bits are needed to encode a planar graph.
Note that this two question are strongly connected. Indeed, if any object of Pn can be encoded with alpha-n bits, then the the size of Pn is no bigger thant 2 power alpha n. If we do not constrain the complexity of the encoding algorithm, the reserve is also true.
The last question we would like to answer is the following: how many edges a typical plane graph has ?

4. Related Works Encoding (n = number of nodes; m = number of edges)
[Turan, 84]: 4m bits
[Keeler, Westbrook, 95]: 3.58m bits
[Munro, Raman, 97]: 2m + 8n bits
[Chiang, Lin, Lu, 01]: 4m/3 + 5n bits
Number of planar graphs
[Osthus, Prömel, Taraz, 02]:  25.22n
[Bousquet-Mélou, 02]:  n
[Bender, Gao, Wormald, 99]:  24.71n
For triangulation [Tutte, 62]: 23.24n
Number of edges (for almost all planar graphs)
[Gerke, McDiamid, 01]: m  2.69n
For labeled [Osthus, Prömel, Taraz, 02]: 1.85n  m  2.56n
Several results gives partial answer to theses questions.
Succinct representation of planar graphshas a long history.
Considering the encoding problem, several algorithms have been proposed with different efficiency.
For the bounds on the enumeration of planar graphs several results have also been given.

5. Encoding Scheme of a Planar Graph
Embed the graph G
Triangulate the graph G
Encode the triangulation
Encode the edges to remove.
Coding size: 3.24n + 3n = 6.24n
Ideas:
Compute a good embedding of G: well-orderly embedding
Compute a good triangulation of G: super-triangulation

6. Realizer (or Schnyder trees) [Schnyder, 89]
R=(T0,T1,T2) is a realizer of a maximal plane graph G if:
T0,T1,T2 make a partition of internal edges of G.
For each internal node v:
Example : v
Thm [Schnyder, 89]: R=(T0,T1,T2) can be computed in linear time

7. Structure of the Realizers of G
cw-triangle
ccw-triangle
Thm [Ossona de Mendez, 94]: The realizers of G is a distributive lattice.
The minimal realizer is the unique realizer of G without any cw-triangle.

8. Super-Triangulation
S=(T0,T1,T2) is a super-triangulation of a planar graph G if:
V(S) = V(G) and E(G)  E(S)
S is a minimal realizer
T0  E(G)
If v is an inner node of T2 then (v, P1(v))  E(G) 1 3 6 2 8 7 5 4 1 3 6 2 8 7 5 4
Thm: Every connected planar graph G has a super-triangulation S that can be computed in O(n) time.

9. Encoding a Planar Graph with a Super-Triangulation
Planar graph = super-triangulation - {missing edges (green + some red)}
Coding: 1

10. Coding a Planar Graph with a Super-Triangulation
Planar graph = super-triangulation - {missing edges (green + some red)}
Decoding: 1

11. Length Coding Analysis
Data structure:
7 binary strings with different density of “1”.
1 binary string for the missing edges.
Each string is compressed considering its density
Thm: Planar graph encoding
5.03n bits (3.37n bits for the super-triangulation)

12. Enumeration Thm: Let p(n) be the number of planar graph with n nodes.
p(n)  n
Thm: For almost all planar graphs, the number of edges m is:
1.70n  m  2.54n
(Previously: ??  m  2.69n )

13. Conclusion Results: Conjecture: p(n)  binomial(5n,2n)  24.85n
An explicit linear time and space algorithm to encode a planar graph with 5.03n bits.
A new upper-bound on the number of planar graphs:
p(n)  n
new bounds on the typical number of edges:
1.70n  m  2.54n
Conjecture: p(n)  binomial(5n,2n)  24.85n
find a better encoding of the super-triangulation
find a better embedding