10 Steps to Counting Unlabeled Planar Graphs: 20 Years Later
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1 10 Steps to : 20 Years Later Manuel Bodirsky October 2007
2 A Sloane Sequence A (core, nice, hard): Number p(n) of unlabeled planar simple graphs with n nodes. Initial terms: 1, 2, 4, 11, 33, 142, 822, 6966, 79853, The Problem
3 A Sloane Sequence A (core, nice, hard): Number p(n) of unlabeled planar simple graphs with n nodes. Initial terms: 1, 2, 4, 11, 33, 142, 822, 6966, 79853, For comparison: number of all unlabeled graphs with n nodes 1, 2, 4, 11, 34, 156, 1044, 12346, , The Problem
4 Unlabeled Enumeration Consider graphs up to isomorphism. For general graphs: the number of labeled and the number of unlabeled graphs are asymptotically equal, since almost all graphs are asymetric. For planar graphs: the number of labeled graphs is much larger than the number of unlabeled graphs, since almost all planar graphs have a large automorphisms group. Tools for unlabeled enumeration: 1 ordinary generating functions 2 Burnside s lemma (orbit counting lemma) 3 cycle indices The Problem
5 Related Tasks Problems related to the enumeration of planar graphs A Compute p(n) in polynomial time in n. B Sample a random planar graph on n vertices in polynomial time in n from the uniform distribution. C Determine the asymptotic growth of p(n). D Devise a Boltzman sampler for random planar graphs. E Analyse properties of random planar graphs In our setting, all these tasks are closely related. The Problem
6 Liskovets and Walsh 1987: Ten Steps V. A. Liskovets, T. R. Walsh: Ten steps to counting planar graphs, Congressus Numerantium (1987). One well-known long-standing unsolved graphenumeration problem is to count (non-isomorphic) planar graphs. The aim of this brief survey is to draw the reader s attention to the considerable progress which has been achieved to that end, and which suggests that this problem may soon be completely solved. The Problem
7 Liskovets and Walsh 1987: Ten Steps V. A. Liskovets, T. R. Walsh: Ten steps to counting planar graphs, Congressus Numerantium (1987). One well-known long-standing unsolved graphenumeration problem is to count (non-isomorphic) planar graphs. The aim of this brief survey is to draw the reader s attention to the considerable progress which has been achieved to that end, and which suggests that this problem may soon be completely solved. The problem is still open. The Problem
8 General Approach Essentially, there is no alternative to the following basic approach: Component Structure Block Decomposition Network Decomposition planar graphs connected planar graphs 2-connected planar graphs 3-connected planar graphs Whitney's theorem: Geometry For the labeled case, this approach has been successful: for exact numbers and random generation (B.,Gröpl,Kang 03) for the asymptotic growth (Gimenez,Noy 05) for Boltzmann generation (Fusy 05) Approach
9 Planar Maps How to count 3-connected planar maps, i.e., 3-connected plane graphs on the sphere, up to homeomorphisms? A question that is credited to Euler polyhedra Approach
10 Planar Maps How to count 3-connected planar maps, i.e., 3-connected plane graphs on the sphere, up to homeomorphisms? A question that is credited to Euler polyhedra Tutte: First count rooted maps, (and try to get rid of the root later...) Approach
11 Planar Maps How to count 3-connected planar maps, i.e., 3-connected plane graphs on the sphere, up to homeomorphisms? A question that is credited to Euler polyhedra Tutte: First count rooted maps, (and try to get rid of the root later...) Go from lower connectivity to higher connectivity Approach
12 Ten Steps symmetry type maps non-sep. maps 3-conn. maps rooted Tutte 63 Tutte 63 Tutte 63 sense-pres. iso. Liskovets 82 Liskovets,Walsh 83 Walsh 82 all map-iso. Wormald 81 Wormald xx Wormald xx Approach
13 Ten Steps symmetry type maps non-sep. maps 3-conn. maps rooted Tutte 63 Tutte 63 Tutte 63 sense-pres. iso. Liskovets 82 Liskovets,Walsh 83 Walsh 82 all map-iso. Wormald 81 Wormald xx Wormald xx Last step from 3-connected to planar graphs: recursive scheme Approach
14 Ten Steps symmetry type maps non-sep. maps 3-conn. maps rooted Tutte 63 Tutte 63 Tutte 63 sense-pres. iso. Liskovets 82 Liskovets,Walsh 83 Walsh 82 Fusy 05 all map-iso. Wormald 81 Wormald xx Wormald xx Last step from 3-connected to planar graphs: recursive scheme Approach
15 Ten Steps symmetry type maps non-sep. maps 3-conn. maps rooted Tutte 63 Tutte 63 Tutte 63 sense-pres. iso. Liskovets 82 Liskovets,Walsh 83 Walsh 82 Fusy 05 all map-iso. Wormald 81 Wormald xx Wormald xx Last step from 3-connected to planar graphs: recursive scheme Cubic Planar Graphs: symmetry type maps non-sep. maps 3-conn. maps rooted Mullin 66 Mullin 65 Tutte 64 sense-pres. iso. Liskovets,Walsh 87 (Brown 64) all map-iso. Tutte 80 Approach
16 The Orbit-counting Lemma Aka Cauchy-Frobenius, or Burnside Lemma. G finite group acting on a set X. The number of orbits of G is 1/ G g G Fix(g) In our setting, for 3-connected planar graphs: X: 3-connected labeled planar graphs with vertices {1,..., n} G = {id,reflections,rotations,reflection-rotations} Orbits of G: unlabeled 3-connected planar graphs Approach
17 Quotient Maps To count graphs with a rotative symmetry of order k, use concept of quotient maps (Liskovets 82,Walsh 82,Fusy 05). Example for k = 3 where both poles of the rotation are faces: Approach
18 Quotient Maps To count graphs with a rotative symmetry of order k, use concept of quotient maps (Liskovets 82,Walsh 82,Fusy 05). Example for k = 3 where both poles of the rotation are faces: Approach
19 Quotient Maps To count graphs with a rotative symmetry of order k, use concept of quotient maps (Liskovets 82,Walsh 82,Fusy 05). Example for k = 3 where both poles of the rotation are faces: Approach
20 Quotient Maps To count graphs with a rotative symmetry of order k, use concept of quotient maps (Liskovets 82,Walsh 82,Fusy 05). Example for k = 3 where both poles of the rotation are faces: Approach
21 Quotient Maps To count graphs with a rotative symmetry of order k, use concept of quotient maps (Liskovets 82,Walsh 82,Fusy 05). Example for k = 3 where both poles of the rotation are faces: Approach
22 Quotient Maps To count graphs with a rotative symmetry of order k, use concept of quotient maps (Liskovets 82,Walsh 82,Fusy 05). Example for k = 3 where both poles of the rotation are faces: Approach
23 Quotient Maps To count graphs with a rotative symmetry of order k, use concept of quotient maps (Liskovets 82,Walsh 82,Fusy 05). Example for k = 3 where both poles of the rotation are faces: Approach
24 Quotient Maps To count graphs with a rotative symmetry of order k, use concept of quotient maps (Liskovets 82,Walsh 82,Fusy 05). Example for k = 3 where both poles of the rotation are faces: Obtain a unique map with two distinguished faces Can be further decomposed (e.g. by using quadrangulations as in Fusy 04) Approach
25 Maps with a Reflective Symmetry Approach
26 Maps with a Reflective Symmetry Several Decompositions and Algorithms: Wormald xx (unpublished algorithm) Approach
27 Maps with a Reflective Symmetry Several Decompositions and Algorithms: Wormald xx (unpublished algorithm) Qadrangulation method in Fusy 05 can in principle be applied here as well Approach
28 Maps with a Reflective Symmetry Several Decompositions and Algorithms: Wormald xx (unpublished algorithm) Qadrangulation method in Fusy 05 can in principle be applied here as well B.,Groepl,Kang 05: Colored connectivity decomposition None of the approaches lead to reasonable formulas so far Approach
29 Colored Decomposition Assume that there is a distinguished directed edge on the symmetry (an arc-root). Resulting graph is 2-connected, and can be decomposed easily. But: have two parameters for number of red and blue vertices Approach
30 Tutte-like Decomposition Similarly to the decomposition of triangulations (Tutte ) and c-nets (B.,Groepl,Johannsen,Kang 05) Approach
31 Tutte-like Decomposition Similarly to the decomposition of triangulations (Tutte ) and c-nets (B.,Groepl,Johannsen,Kang 05) Approach
32 Tutte-like Decomposition Similarly to the decomposition of triangulations (Tutte ) and c-nets (B.,Groepl,Johannsen,Kang 05) Approach
33 Tutte-like Decomposition Similarly to the decomposition of triangulations (Tutte ) and c-nets (B.,Groepl,Johannsen,Kang 05) Approach
34 Tutte-like Decomposition Similarly to the decomposition of triangulations (Tutte ) and c-nets (B.,Groepl,Johannsen,Kang 05) Approach
35 Tutte-like Decomposition Similarly to the decomposition of triangulations (Tutte ) and c-nets (B.,Groepl,Johannsen,Kang 05) Approach
36 Tutte-like Decomposition Similarly to the decomposition of triangulations (Tutte ) and c-nets (B.,Groepl,Johannsen,Kang 05) Approach
37 Tutte-like Decomposition Similarly to the decomposition of triangulations (Tutte ) and c-nets (B.,Groepl,Johannsen,Kang 05) Approach
38 Tutte-like Decomposition Similarly to the decomposition of triangulations (Tutte ) and c-nets (B.,Groepl,Johannsen,Kang 05) Approach
39 Tutte-like Decomposition Similarly to the decomposition of triangulations (Tutte ) and c-nets (B.,Groepl,Johannsen,Kang 05) Advantage: only one extra variable, simple GF equations. But: tedious Approach
40 Another Line of Research Can make progress already before we solve Euler s problem. Graph Class Forbidden Minors Connectivity Structure Planar K 5, K 3,3 Whitney for 3-conn. Easier Problems
41 Another Line of Research Can make progress already before we solve Euler s problem. Graph Class Forbidden Minors Connectivity Structure Planar K 5, K 3,3 Whitney for 3-conn. Series-parallel K 4 No 3-conn. comp. Easier Problems
42 Another Line of Research Can make progress already before we solve Euler s problem. Graph Class Forbidden Minors Connectivity Structure Planar K 5, K 3,3 Whitney for 3-conn. Series-parallel K 4 No 3-conn. comp. Outerplanar K 4, K 2,3 Hamiltonian 2-conn comp. Easier Problems
43 Another Line of Research Can make progress already before we solve Euler s problem. Graph Class Forbidden Minors Connectivity Structure Planar K 5, K 3,3 Whitney for 3-conn. Series-parallel K 4 No 3-conn. comp. Outerplanar K 4, K 2,3 Hamiltonian 2-conn comp. Forest K 3 No 2-conn. comp. Easier Problems
44 Another Line of Research Can make progress already before we solve Euler s problem. Graph Class Forbidden Minors Connectivity Structure Planar K 5, K 3,3 Whitney for 3-conn. Series-parallel K 4 No 3-conn. comp. Outerplanar K 4, K 2,3 Hamiltonian 2-conn comp. Forest K 3 No 2-conn. comp. Results: Graph Class Labeled Unlabeled Planar Series-parallel Outerplanar Forest Well-known Well-known Easier Problems
45 Another Line of Research Can make progress already before we solve Euler s problem. Graph Class Forbidden Minors Connectivity Structure Planar K 5, K 3,3 Whitney for 3-conn. Series-parallel K 4 No 3-conn. comp. Outerplanar K 4, K 2,3 Hamiltonian 2-conn comp. Forest K 3 No 2-conn. comp. Results: Graph Class Labeled Unlabeled Planar Gimenez,Noy 05 Series-parallel Outerplanar Forest Well-known Well-known Easier Problems
46 Another Line of Research Can make progress already before we solve Euler s problem. Graph Class Forbidden Minors Connectivity Structure Planar K 5, K 3,3 Whitney for 3-conn. Series-parallel K 4 No 3-conn. comp. Outerplanar K 4, K 2,3 Hamiltonian 2-conn comp. Forest K 3 No 2-conn. comp. Results: Graph Class Labeled Unlabeled Planar Gimenez,Noy 05 Series-parallel B.,Kang,Gimenez,Noy 05 Outerplanar B.,Kang,Gimenez,Noy 05 Forest Well-known Well-known Easier Problems
47 Another Line of Research Can make progress already before we solve Euler s problem. Graph Class Forbidden Minors Connectivity Structure Planar K 5, K 3,3 Whitney for 3-conn. Series-parallel K 4 No 3-conn. comp. Outerplanar K 4, K 2,3 Hamiltonian 2-conn comp. Forest K 3 No 2-conn. comp. Results: Graph Class Labeled Unlabeled Planar Gimenez,Noy 05? Series-parallel B.,Kang,Gimenez,Noy 05? Outerplanar B.,Kang,Gimenez,Noy 05 B.,Fusy,Kang,Vigerske 07 Forest Well-known Well-known Easier Problems
48 Cycle Indices Polya theory. Let G be a graph with vertices {1,..., n}. Z (G; s 1, s 2,... ) := 1/ Aut(G) n g Aut(G) k=1 where j k (g) is the number of cycles of length k in g. Let K be a class of graphs. Z (K; s 1, s 2,... ) := G K Z (G; s 1, s 2,... ) s j k k (g) Easier Problems
49 2-Connected Outerplanar Graphs Cycle index sum for 2-connected outerplanar graphs Easier Problems
50 2-Connected Outerplanar Graphs Cycle index sum for 2-connected outerplanar graphs Z (D) = 1 2 d 1 ϕ (d) d ( 3 log s d + 1 ) sd 2 4 6s d s 2 + s1 2 4s s2 1 3s2 1 s 2 + 2s 1 s s2 2 1 ( 1 + s s s ) 1 s2 2 s 6s 2 + 1, 2 where ϕ is the Euler-ϕ-function ϕ(n) = n p n (1 p 1 ) + 3 s 1 s s Easier Problems
51 From 2-Connected to Connected Technique be Norman 54, Robinson 70, Harary,Palmer 73 Tool 1: composition Z (G)[Z (K)] := Z (G; Z (K; s 1, s 2,... ), Z (K; s 2, s 4,... ),... ) Tool 2: rooting Tool 3: unrooting Z ( ^G) = s 1 Z (G) s 1 Z (G) = s1 0 1 t 1 Z ( ^G) s1 =t 1 dt 1 + Z (G) s1 =0 Easier Problems
52 From 2-Connected to Connected ^D: cycle index sum for rooted two-connected outerplanar graphs. ^C: cycle index sum for rooted connected outerplanar graphs. Z ( ^C) = s 1 exp Z ( ^D; Z ( ^C; s k, s 2k,... ), Z ( ^C; s 2k, s 4k,... )) k Z ( ^C; s k 1 k, s 2k,... ) The cycle index sum for connected outerplanar graphs Z (C) = Z ( ^C) + Z ( D; Z ( ^C) ) Z ( ^D; Z ( ^C) ) Substituting x i for s i gives equations for the generating functions and a polynomial-time algorithm to compute the numbers. Easier Problems
53 Asymptotic Results With singularity analysis (Flajolet,Sedgewick 0x) we get Theorem 1 (B.,Fusy,Kang,Vigerske 07). The numbers d n, c n, and g n of two-connected, connected, and general outerplanar graphs with n vertices have the asymptotic estimates d n d n 5/2 δ n, c n c n 5/2 ρ n, g n g n 5/2 ρ n, with exponential growth rates δ 1 = and ρ , and constants d , c , and g Easier Problems
54 Cubic Planar Graphs All vertices of degree three. Easier Problems
55 Cubic Planar Graphs All vertices of degree three. Analysed for the labeled case in B.,McDiarmid,Loeffler,Kang 07. Easier Problems
56 Cubic Planar Graphs All vertices of degree three. Analysed for the labeled case in B.,McDiarmid,Loeffler,Kang 07. Why interesting? Easier Problems
57 Cubic Planar Graphs All vertices of degree three. Analysed for the labeled case in B.,McDiarmid,Loeffler,Kang 07. Why interesting? More difficult than SP graphs in that we have to deal with 3-connected components Easier Problems
58 Cubic Planar Graphs All vertices of degree three. Analysed for the labeled case in B.,McDiarmid,Loeffler,Kang 07. Why interesting? More difficult than SP graphs in that we have to deal with 3-connected components However, the number of 3-connected cubic planar graphs is well-understood (bijective correspondence to triangulations) Easier Problems
59 Cubic Planar Graphs All vertices of degree three. Analysed for the labeled case in B.,McDiarmid,Loeffler,Kang 07. Why interesting? More difficult than SP graphs in that we have to deal with 3-connected components However, the number of 3-connected cubic planar graphs is well-understood (bijective correspondence to triangulations) 2-connected and connected numbers are closely related Easier Problems
60 Ten Steps Ten New Steps
61 Ten Steps 1 2-connected, connected, and general series-parallel graphs Ten New Steps
62 Ten Steps 1 2-connected, connected, and general series-parallel graphs 2 2-connected, connected, and general cubic planar graphs Ten New Steps
63 Ten Steps 1 2-connected, connected, and general series-parallel graphs 2 2-connected, connected, and general cubic planar graphs 3 Arc-rooted 3-connected planar maps with a sense-reversing automorphism Ten New Steps
64 Ten Steps 1 2-connected, connected, and general series-parallel graphs 2 2-connected, connected, and general cubic planar graphs 3 Arc-rooted 3-connected planar maps with a sense-reversing automorphism 4 3-connected planar graphs with a sense-reversing automorphism Ten New Steps
65 Ten Steps 1 2-connected, connected, and general series-parallel graphs 2 2-connected, connected, and general cubic planar graphs 3 Arc-rooted 3-connected planar maps with a sense-reversing automorphism 4 3-connected planar graphs with a sense-reversing automorphism 5 3-connected planar graphs with a sense-reversing rotation Ten New Steps
66 Ten Steps 1 2-connected, connected, and general series-parallel graphs 2 2-connected, connected, and general cubic planar graphs 3 Arc-rooted 3-connected planar maps with a sense-reversing automorphism 4 3-connected planar graphs with a sense-reversing automorphism 5 3-connected planar graphs with a sense-reversing rotation 6 Polyhedra (3-connected planar graphs) Ten New Steps
67 Ten Steps 1 2-connected, connected, and general series-parallel graphs 2 2-connected, connected, and general cubic planar graphs 3 Arc-rooted 3-connected planar maps with a sense-reversing automorphism 4 3-connected planar graphs with a sense-reversing automorphism 5 3-connected planar graphs with a sense-reversing rotation 6 Polyhedra (3-connected planar graphs) 7 Arc-rooted 2-connected planar graphs Ten New Steps
68 Ten Steps 1 2-connected, connected, and general series-parallel graphs 2 2-connected, connected, and general cubic planar graphs 3 Arc-rooted 3-connected planar maps with a sense-reversing automorphism 4 3-connected planar graphs with a sense-reversing automorphism 5 3-connected planar graphs with a sense-reversing rotation 6 Polyhedra (3-connected planar graphs) 7 Arc-rooted 2-connected planar graphs 8 2-connected planar graphs Ten New Steps
69 Ten Steps 1 2-connected, connected, and general series-parallel graphs 2 2-connected, connected, and general cubic planar graphs 3 Arc-rooted 3-connected planar maps with a sense-reversing automorphism 4 3-connected planar graphs with a sense-reversing automorphism 5 3-connected planar graphs with a sense-reversing rotation 6 Polyhedra (3-connected planar graphs) 7 Arc-rooted 2-connected planar graphs 8 2-connected planar graphs 9 Connected planar graphs Ten New Steps
70 Ten Steps 1 2-connected, connected, and general series-parallel graphs 2 2-connected, connected, and general cubic planar graphs 3 Arc-rooted 3-connected planar maps with a sense-reversing automorphism 4 3-connected planar graphs with a sense-reversing automorphism 5 3-connected planar graphs with a sense-reversing rotation 6 Polyhedra (3-connected planar graphs) 7 Arc-rooted 2-connected planar graphs 8 2-connected planar graphs 9 Connected planar graphs 10 Planar graphs Ten New Steps
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