Graph classes with given 3 -connected components: asymptotic - PowerPoint PPT Presentation

Background Enumeration Largest block 3-connected component Graph classes with given 3 -connected components: asymptotic enumeration and random graphs Juanjo Ru´ e (joint work with Omer Gim´ enez and Marc Noy) Laboratoire d’Informatique , ´ Ecole Polytechnique , Workshop on Random Graphs and Maps on Surfaces, IHP, 2nd November 2009

Background Enumeration Largest block 3-connected component Background and definitions

Background Enumeration Largest block 3-connected component Objects: graphs Labelled Graph = vertices+edges 3 5 2 4 1 Simple Graph = NO multiples edges, NO loops

Background Enumeration Largest block 3-connected component Objects: graphs crucial concept: graph minor ; tools: connectivy K 4 is a minor of this graph connected → 2 -connected 2 -connected → 3 -connected

Background Enumeration Largest block 3-connected component Generating functions For a combinatorial class A we construct an ordinary generating function (OGF) A ( x ) such that ∞ � a n x n . A ( x ) = n =0 For labelled families of graphs we use exponential GFs (EGF): ∞ x n � n ! y m , A ( x, y ) = a n,m n =0 where x marks vertices and y marks edges .

Background Enumeration Largest block 3-connected component The Symbolic Method COMBINATORIAL RELATIONS between CLASSES ��� EQUATIONS between GENERATING FUNCTIONS Construction OGF EGF A ∪ B Union A ( x ) + B ( x ) A ( x ) + B ( x ) Product A × B A ( x ) · B ( x ) − Labelled Product A ∗ B − A ( x ) · B ( x ) 1 1 Sequence Seq ( A ) 1 − A ( x ) 1 − A ( x ) Set ( A ) − Set exp ( A ( x )) x ∂ x ∂ A • Pointing ∂x A ( x ) ∂x A ( x ) Substitution A ◦ B A ( B ( x )) A ( B ( x ))

Background Enumeration Largest block 3-connected component 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: subexponetial 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)) [ z n ] A ( z ) =

Background Enumeration Largest block 3-connected component 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: subexponetial 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)) [ z n ] A ( z ) =

Background Enumeration Largest block 3-connected component 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: subexponetial 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)) [ z n ] A ( z ) =

Background Enumeration Largest block 3-connected component 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 describes 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 We deal with parameters where these results do not apply. ��� Composition Schemes

Background Enumeration Largest block 3-connected component 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 describes 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 We deal with parameters where these results do not apply. ��� Composition Schemes

Background Enumeration Largest block 3-connected component Definitions, enumeration and normal limit laws

Background Enumeration Largest block 3-connected component Our starting point Asymptotic enumeration and limit laws of planar graphs (Gim´ enez, Noy) g 1 · n − 7 / 2 · γ n 1 · n ! Asymptotic enumeration and limit laws of series-parallel graphs (Bodirsky, Gim´ enez, Kang, Noy) g 2 · n − 5 / 2 · γ n 2 · n !

Background Enumeration Largest block 3-connected component Our starting point Asymptotic enumeration and limit laws of planar graphs [Gim´ enez, Noy] g 1 · n − 7 / 2 · γ n 1 · n ! Asymptotic enumeration and limit laws of series-parallel graphs [Bodirsky, Gim´ enez, Kang, Noy] g 2 · n − 5 / 2 · γ n 2 · n !

Background Enumeration Largest block 3-connected component Our starting point g 1 · n − 7 / 2 · γ n 1 · n ! g 2 · n − 5 / 2 · γ n 2 · n ! � THE SUBEXPONENTIAL TERM GIVES THE “PHYSICS” OF THE GRAPHS � GENERAL FRAMEWORK TO UNDERSTAND THIS EXPONENT

Background Enumeration Largest block 3-connected component Our starting point g 1 · n − 7 / 2 · γ n 1 · n ! g 2 · n − 5 / 2 · γ n 2 · n ! � THE SUBEXPONENTIAL TERM GIVES THE “PHYSICS” OF THE GRAPHS � GENERAL FRAMEWORK TO UNDERSTAND THIS EXPONENT

Background Enumeration Largest block 3-connected component The strategy networks conn. GENERAL 3 − conn. 2 − conn. Planar graphs: 3 − connected planar graphs We get the equations in the opposite way

Background Enumeration Largest block 3-connected component The strategy networks conn. GENERAL 3 − conn. 2 − conn. Series-parallel graphs: − We get the equations in the opposite way

Background Enumeration Largest block 3-connected component The strategy networks conn. GENERAL 3 − conn. 2 − conn. A family of 3-connected graphs We get the equations in the opposite way

Background Enumeration Largest block 3-connected component 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 ))

Background Enumeration Largest block 3-connected component 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 ))

Background Enumeration Largest block 3-connected component 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.

Background Enumeration Largest block 3-connected component 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.

Background Enumeration Largest block 3-connected component 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 )

Background Enumeration Largest block 3-connected component 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 2(1 + y ) ∂B ∂y ( x, y ) = x 2 (1 + D ( x, y )) D is the GF for networks (essentially edge-rooted 2-connected graphs without the root and distinguished end-vertices).

Background Enumeration Largest block 3-connected component 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 2(1 + y ) ∂B ∂y ( x, y ) = x 2 (1 + D ( x, y )) D is the GF for networks (essentially edge-rooted 2-connected graphs without the root and distinguished end-vertices).

Background Enumeration Largest block 3-connected component A set of equations INPUT: T ( x, z ) � 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 ))

Background Enumeration Largest block 3-connected component Examples of families & excluded minors (I) • Series-parallel graphs • Excluded minors: • T : None. • T ( x, z ) = 0 . • Planar graphs • Excluded minors: • T : 3-connected planar graphs. • T ( x, z ) : The number of labelled 2-connected planar graphs (Bender, Gao, Wormald, 2002)