Asymptotic study of subcritical graph classes Michael Drmota, Eric Fusy, Mihyun Kang, Veronika Kraus, Juanjo Ru´ e Laboratoire d’Informatique, ´ Ecole Polytechnique, ERC Exploremaps Project VII Jornadas de Matem´ atica Discreta y Algor´ ıtmica, Castro Urdiales, 7 Julio 2010
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.
Background
Objects: graphs Labelled Graph = labelled vertices+edges . Unlabelled Graph = labelled one up to permutation of labels . Simple Graph = NO multiples edges, NO loops . 2 1 3 1 1 2 3 2 3 2 3 1 Question : How many graphs with n vertices are in the family?
The counting series Strategy : Encapsulate these numbers → Counting series ◮ Labelled framework: exponential generating functions ∞ ∑ ∑ x | a | |A n | n ! x n A ( x ) = | a | ! = a ∈A n ≥ 0 ◮ Unlabelled framework: cycle index sums ∑ ∑ 1 s c 1 1 s c 2 2 · · · s c n Z A ( s 1 , s 2 , . . . ) = n , n ! n ≥ 0 ( σ,g ) ∈ S n ×A n σ · g = g ∑ � | � A ( x ) = Z A ( x, x 2 , x 3 , . . . ) = A n | x n . n ≥ 0
The symbolic method COMBINATORIAL RELATIONS between CLASSES ↕⇕↕ EQUATIONS between GENERATING FUNCTIONS Class Labelled setting Unlabelled setting C ( x ) = � � A ( x ) + � C = A ∪ B C ( x ) = A ( x ) + B ( x ) B ( x ) C ( x ) = � � A ( x ) · � C = A × B C ( x ) = A ( x ) · B ( x ) B ( x ) ( ∑ ) � i � 1 B ( x i ) C = Set( B ) C ( z ) = exp( B ( x )) C ( x ) = exp i ≥ 1 C ( x ) = Z A ( � � B ( x ) , � B ( x 2 ) , . . . ) C = A ◦ B C ( x ) = A ( B ( x ))
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 Γ( α ) · n α − 1 · ρ − n (1 + o (1)) a n = [ z n ] A ( z ) ∼
Limit laws Study of parameters → A ( u, z ) = ∑ ∞ n,m =0 a n,m z n u m . For a fixed n , the numbers a n,m describe a discrete probability law X n = [ u m z n ] A ( u, z ) a n,m p ( X n = m ) = ∑ ∞ [ z n ] A (1 , z ) m =0 a n,m Does X n converge in distribution to a random variable X? We expect normal limit distributions: general theorems
Families of graphs under study
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 • ))
Some families of graphs (1) 1 . Plane trees ( Ex( K 3 ) ): 1 . Explicit expressions
Some families of graphs (2) 2 . Cacti graphs : 1 . Explicit expressions s 2 1 + s 1 1 Z B′ ( s 1 , s 2 , . . . ) = s 1 + + , 2(1 − s 1 ) 2(1 − s 2 )
Some families of graphs (3) 3 . Outerplanar graphs ( Ex( K 4 , K 2 , 3 ) ): 1 . Explicit expressions (Bodirsky, Fusy, Kang, Vigerske, 2007) ( 3 s 2 + s 2 1 ϕ ( d ) 1 1 1 − 4 s 1 − 2 ) √ ∑ s 2 Z B ( s 1 , . . . ) = log s d + d − 6 s d + 1 + − − 2 d 4 4 4 16 d> 0 s 2 1 − 3 s 2 s 2 ( ) √ 1 s 2 + 2 s 1 s 2 3 − s 1 1 s 1 √ 1 s 2 s 2 + + 1 − 6 s 1 + 1 − 1 + + 2 2 − 6 s 2 + 1 16 s 2 s 2 16 16 s 2 2 2
Some families of graphs (and 4) 4 . Series-parallel graphs (Ex( K 4 )) : 5 . NO explicit expressions !
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 ))) , (∑ ) � i Z B ′ ( � C • ( x i ) , � 1 C • ( x 2 i ) , . . . ) C • ( x ) = x exp . 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.
Graph decomposition, a grammar and system of functional equations
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 ))
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 ))
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. C o = v × Set( B o ( v ← C o )) = ⇒ xC ′ ( x ) = x exp B ′ ( xC ′ ( x ))
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. C • = v × SET( B o ( v ← C • )) = ⇒ xC ′ ( x ) = x exp B ′ ( xC ′ ( x ))
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 )
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 ) 1 ∂T D ( x, y ) = (1 + y ) exp 1 + xD + ∂z ( x, D ) − 1 2 x 2 ( 1 + D ( x, y ) ) ∂y ( x, y ) = x 2 ∂B 2 1 + y D is the GF for networks (essentially edge-rooted 2-connected graphs without the edge root).
A set of equations ( 1 + D ) xD 2 1 ∂T ∂z ( x, D ) − log + 1 + xD = 0 2 x 2 D 1 + y ( 1 + D ( x, y ) ) ∂y ( x, y ) = x 2 ∂B 2 1 + y ( ) C • ( x, y ) = exp B ′ ( C • ( x, y ) , y ) G ( x, y ) = exp( C ( x, y ))
But in the unlabelled framework, things are more involved...
The complete grammar A Grammar for Decomposing a Family of Graphs into 3-connected Components; Chapuy, Fusy, Kang, Shoilekova This system is obtained ap- plying the dissymmetry theorem for trees in an ingenious way . Hence, in the unlabelled framework we need to study system of functional equations more involved.
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)) .
Results and explicit values
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.
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 √ V ar X n → N (0 , 1) , Problem : it is usual that we do not know to prove that V ar X n ̸ = 0 without explicit computation ⇓ We find a general analytic criteria on the counting series which assures that V ar X n ̸ = 0.
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.
A numerical table Constant growth for different subcritical graph families Family Labelled Unlabelled Acyclic 2 , 71828 2 , 95577 Cacti 4 , 18865 4 , 50144 Outerplanar 7 , 32708 7 , 50360 Series-Parallel 9 , 07359 9.38527 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)
Open problems
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