Applications of Tuttes tree decomposition in the enumeration of - PowerPoint PPT Presentation

Applications of Tutte’s tree decomposition in the enumeration of bipartite graph families Juanjo Ru´ e , Kerstin Weller Nice Random Graphs Workshop

Objects: graphs Labelled Graph = vertices+edges 3 5 2 4 1 Simple Graph = NO multiples edges, NO loops A family G under study must be: STABLE (closed under permutation of the labels) CLOSED under Tutte’s decomposition ( G ∈ G if and only if its 3-connected components are in G ) PROTOTYPE: planar graphs ← SP-graphs

Contents ◮ Connectivity and enumeration ◮ A combinatorial trick: The Dissymmetry Theorem for trees ◮ The Ising model and the equations for bipartite Series-Parallel graphs

Connectivity and enumeration

The Symbolic Method ` a la Flajolet COMBINATORIAL RELATIONS between CLASSES ↕⇕↕ EQUATIONS between GENERATING FUNCTIONS Construction EGF Union A ∪ B A ( x ) + B ( x ) Labelled Product A × B A ( x ) · B ( x ) 1 Sequence Seq ( A ) 1 − A ( x ) Set Set ( A ) exp ( A ( x )) A • \ \ A ◦ x ∂ ∂x A ( x ) \ \ ∂ Pointing ∂x A ( x ) Substitution A ◦ B A ( B ( x )) GF will be exponential in x (vertices), and ordinary in y (edges) x n ∑ n ! y m a n,m n,m ≥ 0

The Strategy General graph Connected graph 2-connected graph Networks Map Steinitz's 3-connected World graph Theorem For SP - graphs, there are NOT 3-connected components

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 A vertex-rooted connected graph is a tree of blocks. C • = v × Set( B ◦ ( v ← C • )) = ⇒ xC ′ ( x, y ) = x exp B ′ ( xC ′ ( x, y ) , y )

2-connected graphs from 3-connected graphs We study networks : graphs with two privileged virtual vertices ( poles ) such that adding and edge between them makes it 2-connected 2(1 + y ) ∂B ∂y ( x, y ) = x 2 (1 + D ( x, y )) 3 types: SERIES , PARALLEL and T - composition D ( x, y ) = y + S ( x, y ) + P ( x, y ) + H ( x, y ) S ( x, y ) = ( D ( x, y ) − S ( x, y )) xD ( x, y ) P ( x, y ) = y exp ≥ 1 ( S ( x, y ) + H ( x, y )) + exp ≥ 2 ( S ( x, y ) + H ( x, y )) H ( x, y ) = 2 ∂ ∂z T ( x, D ( x, y )) x 2

Simplifying a little... Eliminating variables: ( xD 2 1 ) ∂ − 1 D = (1 + y ) exp 1 + xD + ∂z T ( x, D ) 2 x 2 ∂y = x 2 ∂B 1 + D 2 1 + y

A set of equations INPUT: T ( x, z ) xD 2 1 ( 1 + D )  ∂ ∂z T ( x, D ) − log + 1 + xD = 0  2 x 2  1 + y  ∂y ( x, y ) = x 2 ∂B ( 1 + D )  → INTEGRATION   2 1 + y  xC ′ ( x, y ) = x exp B ′ ( xC ′ ( x, y ) , y )  G ( x, y ) = exp( C ( x, y ))  To have in mind : this integration step could become VERY difficult if we consider enriched families of graphs.

The integration step We have this integral: ∫ y ∂y ( x, y ) = x 2 → B ( x, y ) = x 2 ∂B ( 1 + D ) 1 + D ( x, s ) ds 2 1 + y 2 1 + s 0 Surprisingly, we get an EXACT expression: T ( x, D ( x, y )) − 1 2 xD + 1 2 log(1 + xD ) + ( 1 + y x 2 ( D + 1 )) 2 D 2 + (1 + D ) log . 2 1 + D Are there combinatorial reasons to get such an easy formula?

A combinatorial trick: The Dissymmetry Theorem for trees

A toy example: trees We apply the previous grammar to count ROOTED trees ⇒ T = • × Set( T ) → T ( x ) = xe T ( x ) To forget the root, we just integrate: ( xU ′ ( x ) = T ( x )) ∫ x ∫ T ( x ) { } T ( s ) T ( s ) = u 1 − u du = T ( x ) − 1 2 T ( x ) 2 ds = = T ′ ( s ) ds = du s 0 T (0)

The Dissymmetry Theorem for trees That can be explained only by combinatorial means: T ∪ T •→• ≃ T •−• ∪ T • where T ∗ is the family of trees T with an extra rooted structure. U ( x ) = U •→• ( x ) + U •−• ( x ) − U •→• ( x ) = 1 2 T ( x ) 2 + T ( x ) − T ( x ) 2 For tree-like families we have an extension of this result: A ∪ A •→• ≃ A •−• ∪ A •

Tutte’s tree-like decomposition blocks admit a tree-like decomposition ` a la Tutte: This is a 2-connected SP-graph.

Tutte’s tree-like decomposition blocks admit a tree-like decomposition ` a la Tutte: We split it using 2-cuts

Tutte’s tree-like decomposition blocks admit a tree-like decomposition ` a la Tutte: R M R R M R M R R M R M R The tree has two type of vertices: M , R

Encoding using GFs Developed by Chapuy, Fusy, Kang, Shoilekova. B ( x, y ) = B R ( x, y ) + B M ( x, y ) − B R − M ( x, y ) B R B B M R-M

The Ising model and the equations for bipartite Series-Parallel graphs

Bipartite graphs Is it possible to apply the same arguments for bipartite graphs? No problem from connected to 2-connected From 2-connected to 3-connected we have problems... We need to study something more complicated: Ising model on graphs .

An statistical model Consider graphs with two types of edges: • − • and ◦ − • , and write the GF x n ∑ n ! y r •−• y s d n,r,s ◦−• . n,r,s ≥ 0 Observe that permuting colors on vertices do NOT change the type of the edges: vertex coloring is an auxiliar tool Let us try to get B ( x, y •−• , y ◦−• ) in terms of certain networks : 2(1 + y •−• ) ∂B ( x, y •−• , y ◦−• ) + 2(1 + y ◦−• ) ∂B ( x, y •−• , y ◦−• ) ∂y •−• ∂y ◦−• = x 2 (1 + D •−• ( x, y •−• , y ◦−• ) + D ◦−• ( x, y •−• , y ◦−• )) ...It is better to integrate this COMBINATORIALLY ...

The networks in the model We distinguish depending on the type of the root: D ◦−• = y ◦−• + S ◦−• + P ◦−• D •−• = y •−• + S •−• + P •−• S •−• = ( D •−• − S •−• ) xS •−• + ( D ◦−• − S ◦−• ) xS ◦−• S ◦−• = ( D •−• − S •−• ) xS ◦−• + ( D ◦−• − S ◦−• ) xS •−• P ◦−• = y ◦−• Set ≥ 1 ( S ◦−• ) + Set ≥ 2 ( S ◦−• ) P •−• = y •−• Set ≥ 1 ( S •−• ) + Set ≥ 2 ( S •−• )

From networks to 2-connected graphs Now we can write B R , B M and B RM , but we have more cases: B R B B M R-M Then we get a long expression in terms of the GF for networks.

Going back to the bipartite model Working a little, we can get two equations relating the bipartite families D ◦−• ( x, y, 0) = D 2 and D •−• ( x, y, 0) = D 1 D 13 x − D 1 D 22 x + D 12 + D 22 ) ( ) ( x D 1 = − 1 + exp D 12 x 2 − D 22 x 2 + 2 xD 1 + 1 xD 12 − xD 22 + 2 D 1 ( ( ) ) ( ) xD 2 D 2 = y + y exp − 1 D 12 x 2 − D 22 x 2 + 2 xD 1 + 1 xD 12 − xD 22 + 2 D 1 ( ) ( ) xD 2 + exp − 1 . D 12 x 2 − D 22 x 2 + 2 xD 1 + 1 And then, we can substitute in the previous expressions for B

Asymptotic enumeration Applying the machinery of analytic combinatorics we have the following theorem [R., Weller, 2014] The number of connected and general bipartite (labelled) series-parallel graphs with n vertices is asymptotically equal to c n ∼ c · n − 5 / 2 · γ n · n ! g n ∼ g · n − 5 / 2 · γ n · n ! where γ ≈ 4 , 22044, c ≈ 0 , 021446 and g ≈ 0 , 026499 are computable constants. The constant growth for SP graphs is γ ′ ≈ 9 , 07359

...And random graphs We have then a precise probability of a random SP-graph being bipartite [R., Weller, 2014] The probability that a uniformly connected random SP-graph with n vertices is bipartite is 0 , 3167 · (2 , 1499) − n (1 + o (1)) We can also obtain limiting distributions for the number of edges, cutvertices, blocks (Normal distribution, linear expectation).

Moltes gr` acies!

Applications of Tutte’s tree decomposition in the enumeration of bipartite graph families Juanjo Ru´ e , Kerstin Weller Nice Random Graphs Workshop