La BRI Universit Bordeaux I STACS 03 An

La. BRI, Université Bordeaux I STACS’ 03 An Information-Theoretic Upper Bound of Planar Graphs Using Triangulation Nicolas Bonichon, Cyril Gavoille & Nicolas Hanusse

Planar Graphs (unlabeled) • Planar graph • Maximal planar graph (or triangulation) • Plane embedding (or plane graph)

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)?

Related Works • Encoding (n = number of nodes; m = number of edges) – – [Turan, 84]: 4 m bits [Keeler, Westbrook, 95]: 3. 58 m bits [Munro, Raman, 97]: 2 m + 8 n bits [Chiang, Lin, Lu, 01]: 4 m/3 + 5 n bits • Number of planar graphs – – [Osthus, Prömel, Taraz, 02]: 25. 22 n [Bousquet-Mélou, 02]: 25. 098 n [Bender, Gao, Wormald, 99]: 24. 71 n For triangulation [Tutte, 62]: 23. 24 n • Number of edges (for almost all planar graphs) – [Gerke, Mc. Diamid, 01]: m 2. 69 n – For labeled [Osthus, Prömel, Taraz, 02]: 1. 85 n m 2. 56 n

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. 24 n + 3 n = 6. 24 n • Ideas: – Compute a good embedding of G: well-orderly embedding – Compute a good triangulation of G: super-triangulation

Realizer (or Schnyder trees) [Schnyder, 89] • R=(T 0, T 1, T 2) is a realizer of a maximal plane graph G if: Example : – T 0, T 1, T 2 make a partition of internal edges of G. – For each internal node v: v Thm [Schnyder, 89]: R=(T 0, T 1, T 2) can be computed in linear time

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.

Super-Triangulation • S=(T 0, T 1, T 2) is a supertriangulation of a planar graph G if: – – V(S) = V(G) and E(G) E(S) S is a minimal realizer T 0 E(G) If v is an inner node of T 2 then (v, P 1(v)) E(G) 6 2 5 8 3 7 4 1 Thm: Every connected planar graph G has a super-triangulation S that can be computed in O(n) time.

Encoding a Planar Graph with a Super-Triangulation • Planar graph = super-triangulation - {missing edges (green + some red)} • Coding: 1 0 0 0 11 0 0 1 1 10011 0110

Coding a Planar Graph with a Super-Triangulation • Planar graph = super-triangulation - {missing edges (green + some red)} • Decoding: 1 0 0 0 11 0 0 1 1 10011 0110

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. 03 n bits (3. 37 n bits for the super-triangulation)

Enumeration Thm: Let p(n) be the number of planar graph with n nodes. p(n) 25. 007 n Thm: For almost all planar graphs, the number of edges m is: 1. 70 n m 2. 54 n (Previously: ? ? m 2. 69 n )

Conclusion • Results: – An explicit linear time and space algorithm to encode a planar graph with 5. 03 n bits. – A new upper-bound on the number of planar graphs: p(n) 25. 007 n – new bounds on the typical number of edges: 1. 70 n m 2. 54 n • Conjecture: p(n) binomial(5 n, 2 n) 24. 85 n – find a better encoding of the super-triangulation – find a better embedding