Asymptotic study of subcritical graph classes

Transcription

1 Asymptotic study of subcritical graph classes Michael Drmota, Eric Fusy, Mihyun Kang, Veronika Kraus, Juanjo Rué Laboratoire d Informatique, École Polytechnique, ERC Exploremaps Project VII Jornadas de Matemática Discreta y Algorítmica, Castro Urdiales, 7 Julio 2010

2 The material of this talk 1. Background and notation. 2. Naive description of the graphs we want to enumerate: subcritical graph families. 3. The strategy: graph decompositions, the grammar and functional system of equations. 4. Results, and explicit computations. 5. Further research and open problems.

3 Background

4 Objects: graphs Labelled Graph= labelled vertices+edges. Unlabelled Graph= labelled one up to permutation of labels. Simple Graph= NO multiples edges, NO loops Question: How many graphs with n vertices are in the family?

5 The counting series Strategy: Encapsulate these numbers Counting series Labelled framework: exponential generating functions A(x) = a A x a a! = n 0 A n x n n! Unlabelled framework: cycle index sums Z A (s 1, s 2,...) = n 0 1 n! (σ,g) S n A n σ g=g Ã(x) = Z A (x, x 2, x 3,...) = n 0 s c 1 1 sc 2 2 scn n, Ãn x n.

6 The symbolic method COMBINATORIAL RELATIONS between CLASSES EQUATIONS between GENERATING FUNCTIONS Class Labelled setting Unlabelled setting C = A B C(x) = A(x) + B(x) C(x) = Ã(x) + B(x) C = A B C(x) = A(x) B(x) C(x) = Ã(x) B(x) C = Set(B) C(z) = exp(b(x)) C(x) = exp ( i 1 1 i B(x i ) ) C = A B C(x) = A(B(x)) C(x) = ZA ( B(x), B(x 2 ),...)

7 Singularity analysis on generating functions GFs: analytic functions in a neighbourhood of the origin. The smallest singularity of A(z) determines the asymptotics of the coefficients of A(z). POSITION: exponential growth ρ. NATURE: subexponential growth Transfer Theorems: Let α / {0, 1, 2,...}. If A(z) = a (1 z/ρ) α + o((1 z/ρ) α ) then a n = [z n ]A(z) a Γ(α) nα 1 ρ n (1 + o(1))

8 Limit laws Study of parameters A(u, z) = n,m=0 a n,mz n u m. For a fixed n, the numbers a n,m describe a discrete probability law X n p(x n = m) = a n,m m=0 a n,m = [um z n ]A(u, z) [z n ]A(1, z) Does X n converge in distribution to a random variable X? We expect normal limit distributions: general theorems

9 Families of graphs under study

10 The main construction We use easier graphs as fundamental pieces. Let B be a family of 2-connected graphs. C : connected graphs with blocks in B. In other words, a vertex-rooted connected graph is a tree of 2-connected blocks. C = v Set(B (v C ))

11 Some families of graphs (1) 1. Plane trees (Ex(K 3 )): 1. Explicit expressions

12 Some families of graphs (2) 2. Cacti graphs: 1. Explicit expressions Z B (s 1, s 2,...) = s 1 + s 2 1 2(1 s 1 ) s 1 2(1 s 2 ),

13 Some families of graphs (3) 3. Outerplanar graphs (Ex(K 4, K 2,3 )): 1. Explicit expressions (Bodirsky, Fusy, Kang, Vigerske, 2007) Z B (s 1,... ) = 1 2 d>0 φ(d) + s2 1 3s2 1 s 2 + 2s 1 s 2 16s 2 2 d ( 3 log s d + 1 ) s 2 d 4 6s d s 2 + s 2 1 4s ( + 3 s 1 s s s2 1 s s 1 2 s 2 ) s 2 2 6s 2 + 1

14 Some families of graphs (and 4) 4. Series-parallel graphs (Ex(K 4 )): 5. NO explicit expressions!

15 The subcritical condition All the previous families are defined in the following way: C = v Set(B (v C )) Which translates into the equations C (x) = x exp(b (C (x))), ( C 1 (x) = x exp i Z B ( C (x i ), C ) (x 2i ),... ). i 1 In both cases, the counting series for connected graphs is determined by the counting series for 2-connected Subcritical condition The singularity for the connected counting series is related to a branch point (derivative equals to 0) of the 2-connected counting series.

16 Graph decomposition, a grammar and system of functional equations

17 General graphs from connected graphs Let C be a family of connected graphs. G : graphs such that their connected components are in C. G = Set(C) = G(x, y) = exp(c(x, y))

18 General graphs from connected graphs Let C be a family of connected graphs. G : graphs such that their connected components are in C. G = Set(C) = G(x, y) = exp(c(x, y))

19 Connected graphs from 2-connected graphs Let B be a family of 2-connected graphs. C : connected graphs with blocks in B. In other words, a vertex-rooted connected graph is a tree of 2-connected blocks.

20 Connected graphs from 2-connected graphs Let B be a family of 2-connected graphs. C : connected graphs with blocks in B. In other words, a vertex-rooted connected graph is a tree of 2-connected blocks.

21 Connected graphs from 2-connected graphs A vertex-rooted connected graph is a tree of rooted blocks. C = v Set(B (v C )) = xc (x, y) = x exp B (xc (x, y), y)

22 2-connected graphs from 3-connected graphs Decomposition in 3-connected components is slightly harder. Let T be a family of 3-connected graphs: T (x, z). We define B as those 2-connected graphs such that can be obtained from series, parallel, and T -compositions. ( xd 2 D(x, y) = (1 + y) exp 1 + xd + 1 T 2x 2 z ( ) 1 + D(x, y) B x2 (x, y) = y y ) (x, D) 1 D is the GF for networks (essentially edge-rooted 2-connected graphs without the edge root).

23 A set of equations 1 T 2x 2 D z B x2 (x, y) = y 2 ( ) 1 + D (x, D) log + xd2 1 + y 1 + xd = 0 ( ) 1 + D(x, y) 1 + y C (x, y) = exp ( B (C (x, y), y) ) G(x, y) = exp(c(x, y))

24 But in the unlabelled framework, things are more involved...

25 The complete grammar A Grammar for Decomposing a Family of Graphs into 3-connected Components; Chapuy, Fusy, Kang, Shoilekova This system is obtained applying the dissymmetry theorem for trees in an ingenious way. Hence, in the unlabelled framework we need to study system of functional equations more involved.

26 A system of functional equations If a combinatorial system of equations is regular enough, we can assure square-root developements [Drmota, 1997] Consider the functional system of equations y = F(y; z, v). If the system satisfies some nice conditions at v = v 0, then, around v = v 0 There is a unique vector of power series y = y(z, v) in the variables z, v that satisfies the system. The components of y have non-negative coefficients [z n ] y i (z, v 0 ) (for i {1,..., r}). The components of y have a square-root expansion around (z 0, v 0 ). Expansions of the form c n 3/2 ρ n (1 + o(1)).

27 Results and explicit values

28 Asymptotic enumeration Our main result is the following one: [Drmota, Fusy, Kang, Kraus R., 2010] Let G be a subcritical block-stable graph class (either labelled or unlabelled). Then, [z n ]C (z) = c 1 n 3/2 γ n (1 + o(1)), [z n ]G(z) = c 2 n 5/2 γ n (1 + o(1)), and for certain constants c 1, c 2, γ. Exponent n 3/2 = arborescent structure= branch point.

29 Limit laws (1) We study parameters on a random connected graph with n vertices: number of edges, number of cut-vertices, number of blocks. In all cases (independently of the framework) we get X n E X n Var Xn N(0, 1), Problem: it is usual that we do not know to prove that Var X n 0 without explicit computation We find a general analytic criteria on the counting series which assures that Var X n 0.

30 Limit laws (2) We study the Degree distribution X k n: number of vertices of degree k in a randomly chosen graph with n vertices. d k the limiting probability that the root vertex of a randomly chosen graph is k. We show the following: We get closed expressions for d k. X k n has a normal limiting distribution.

31 A numerical table Constant growth for different subcritical graph families Family Labelled Unlabelled Acyclic 2, ,95577 Cacti 4, ,50144 Outerplanar 7, ,50360 Series-Parallel 9, The constant growth for unlabelled SP-graphs has been obtained using Generation of the first terms of the counting series. Approximating the system of equations by another easier. Checking convergence of the singular point associated to the system (Pivoteau, Salvy, Soria)

32 Open problems

33 Beyond the subcritical scheme The subcritical condition can be solved easily, compared with a critical condition. Families of graphs which arise from the map context do not satisfy a subcritical condition Next step: take a natural family of 3-connected graphs arising from maps (triangulations), and study the critical scheme.

34 The enumeration of unlabelled planar graphs The problem was completely solved by Giménez and Noy, but little is known in the unlabelled setting: It is necessary to obtain the counting series for 3 connected planar maps (unlabelled): exact enumeration of 3-polytopes. It is necessary to deal with these counting series! It is necessary to solve the critical problem to get the enumeration of vertex-rooted connected planar graphs. Lot of work to be done!

35 Thank you

36 Asymptotic study of subcritical graph classes Michael Drmota, Eric Fusy, Mihyun Kang, Veronika Kraus, Juanjo Rué Laboratoire d Informatique, École Polytechnique, ERC Exploremaps Project VII Jornadas de Matemática Discreta y Algorítmica, Castro Urdiales, 7 Julio 2010