slides - Aofa 2014
Introduction
Properties of graph classes
Counting
More generally
Random graphs from a minor-closed class Colin McDiarmid Oxford University
Analysis of Algorithms, Paris, June 2014
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
1 / 40
Introduction
Properties of graph classes
Counting
More generally
Question
Let A be a class of (simple) graphs closed under isomorphism, eg the class P of planar graphs. An is the set of graphs in A on vertices 1, . . . , n. Rn ∈u A means that Rn is picked uniformly at random from An . What are typical properties of Rn ? usually a giant component? probability of being connected? many vertices of degree 1? size of the 2-core? Can we learn anything useful for the design or analysis of algorithms?
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
2 / 40
Introduction
Properties of graph classes
Counting
More generally
Generating functions
Given P a class A of graph, A(x) denotes the exponential generating function (egf) n |An |x n /n!. Also ρ = ρ(A) = ρ(A) is the radius of convergence. For suitable classes of graphs, we can relate the egfs (or two variable versions) of all graphs, connected graphs, 2-connected graphs and 3-connected graphs. If we know enough about the 3-connected graphs (as we do for planar graphs, thanks to Tutte and others) then we may be able to extend to all graphs. Let us leave that for now and proceed in greater generality.
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
3 / 40
Introduction
Properties of graph classes
Counting
More generally
Minors H is a minor of G if H can be obtained from a subgraph of G by edge-contractions. A is minor-closed if G ∈ A, H a minor of G
⇒ H∈A
Examples: forests, series-parallel graphs, and more generally graphs of treewidth ≤ k; outerplanar graphs, planar graphs, and more generally graphs embeddable on a given surface; graphs with at most k (vertex) disjoint cycles.
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
4 / 40
Introduction
Properties of graph classes
Counting
More generally
Minors
Ex(H) is the class of graphs with no minor a graph in H. For example: series-parallel graphs = Ex(K4 ), planar graphs = Ex({K5 , K3,3 }), graphs with no two disjoint cycles = Ex(2C3 ). Easy to see that: A is minor-closed iff A = Ex(H) for some class H. Robertson and Seymour’s graph minors theorem (once Wagner’s conjecture) is that if A is minor-closed then A = Ex(H) for some finite class H. The unique minimal such H consists of the excluded minors for A.
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
5 / 40
Introduction
Properties of graph classes
Counting
More generally
Minors Mostly we shall assume that A is minor-closed and proper (that is, not empty and not all graphs). For such A, a result of Mader says: there is a c = c(A) such that the average degree of each graph in A is at most c. Thus our √ graphs are sparse. For Ex(Kt ) the maximum average degree is of order t log t (Kostochka, Thomason). Call A small if ρ(A) > 0, that is ∃c such that |An | ≤ c n n! . Norine, Seymour, Thomas and Wollan (2006); and Dvor´ak and Norine (2010) showed that: Each (proper) minor-closed graph class A is small.
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
6 / 40
Introduction
Properties of graph classes
Counting
More generally
Decomposable If a graph is in A if and only if each component is, then we call A decomposable. For example the class of planar graph is decomposable but the class of graphs embeddable on the torus is not. A minor-closed class is decomposable iff each excluded minor is connected.
Let A be a decomposable class of graphs; and let C consist of the connected graphs in A, with egf C (x). The exponential formula says that A(x) = e C (x) .
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
7 / 40
Introduction
Properties of graph classes
Counting
More generally
Bridge-addable and addable
A is bridge-addable if whenever G ∈ A and u and v are in different components of G then G + uv ∈ A. A is addable if it is decomposable and bridge-addable. A minor-closed class A is addable iff each excluded minor is 2-connected. G S is bridge-addable but not decomposable (and so not addable) except in the planar case.
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
8 / 40
Introduction
Properties of graph classes
Counting
More generally
Bridge-addability and being connected
From McD, Steger and Welsh (2005):
Lemma If A is bridge-addable and Rn ∈u A then P(Rn is connected) ≥ 1/e. 1
For trees T and forests F, |Tn | = nn−2 and |Fn | ∼ e 2 nn−2 . Thus for Fn ∈u F, 1 P(Fn is connected) ∼ e − 2 .
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
9 / 40
Introduction
Properties of graph classes
Counting
More generally
Bridge-addability and being connected
McD, Steger and Welsh (2006) conjectured:
Conjecture 1
If A is bridge-addable then P(Rn is connected) ≥ e − 2 +o(1) . Balister, Bollob´as and Gerke (2008, 2010) give an asymptotic lower bound of e −0.7983 . Norine (2013) improves this to e −2/3 , but the full conjecture is still open. Addario-Berry, McD and Reed (2012), and Kang and Panagiotou (2013) prove the conjecture if A is also closed under deleting bridges, that is if A is bridge-alterable.
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
10 / 40
Introduction
Properties of graph classes
Counting
More generally
Bridge-alterability and connectivity
Here is a natural strengthening of the last conjecture, see eg Balister, Bollob´as and Gerke (2010).
Conjecture Let A be bridge-addable, Rn ∈u A and Fn ∈u F. Then P(Rn is connected) ≥ P(Fn is connected). 1
(Recall that P(Fn is connected) ∼ e − 2 .)
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
11 / 40
Introduction
Properties of graph classes
Counting
More generally
Bridge-alterability and connectivity
The result below (2013) gives a weakened form of the conjecture.
Theorem Let A be bridge-alterable, Rn ∈u A, and Ft ∈u F for t = 1, 2, . . .. Then P(Rn is connected) ≥
min P(Ft is connected).
n/3 0 such that Pr (Rn has < αn pendant copies of H) = e −Ω(n) . Often this shows that there are linear numbers of vertices of each degree, and exponentially many automorphisms. For Rn ∈u P, whp ω(Rn ) = 4 and so χ(Rn ) = 4. Hadwiger’s Conjecture being false says that for some k, there is a graph G ∈ Ex(Kk ) with χ(G ) ≥ k. But then for Rn ∈u Ex(Kk ), wvhp χ(Rn ) ≥ k. All but an exponentially small proportion of (Kk ) are counterexamples! Colin McDiarmid (Oxford)
Graphs from a minor-closed class
18 / 40
Introduction
Properties of graph classes
Counting
More generally
Pendant copies theorem Theorem Let A have a finite positive growth constant, and let H be freely attachable to A. Let Rn ∈u A. Then there exists α > 0 such that Pr (Rn has < αn pendant copies of H) = e −Ω(n) . Often this shows that there are linear numbers of vertices of each degree, and exponentially many automorphisms. For Rn ∈u P, whp ω(Rn ) = 4 and so χ(Rn ) = 4. Hadwiger’s Conjecture being false says that for some k, there is a graph G ∈ Ex(Kk ) with χ(G ) ≥ k. But then for Rn ∈u Ex(Kk ), wvhp χ(Rn ) ≥ k. All but an exponentially small proportion of (Kk ) are counterexamples! Colin McDiarmid (Oxford)
Graphs from a minor-closed class
18 / 40
Introduction
Properties of graph classes
Counting
More generally
Pendant copies theorem searching for a subgraph
Let A have a growth constant, and let the connected graph H be freely attachable. For example, let A be addable and minor-closed and let H ∈ A; or let A be G S and H be planar. Suppose we want to find a copy of H in Rn or verify there is no such subgraph. How quickly can we do so? We see that we can do so in constant expected time; and similarly if we can seek an induced copy of H or a minor H.
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
19 / 40
Introduction
Properties of graph classes
Counting
More generally
Pendant copies theorem searching for a subgraph
Let A have a growth constant, and let the connected graph H be freely attachable. For example, let A be addable and minor-closed and let H ∈ A; or let A be G S and H be planar. Suppose we want to find a copy of H in Rn or verify there is no such subgraph. How quickly can we do so? We see that we can do so in constant expected time; and similarly if we can seek an induced copy of H or a minor H.
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
19 / 40
Introduction
Properties of graph classes
Counting
More generally
Smoothness
Let A be any small class of graphs. Call A smooth if
|An | n|An−1 |
→ a limit as n → ∞.
In this case the limit must be the growth constant γ. All the classes for which we know an asymptotic counting formula are smooth, for example series-parallel graphs, P, G S . Showing smoothness is an important step in proving results about Rn ∈u A.
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
20 / 40
Introduction
Properties of graph classes
Counting
More generally
When is A smooth? Bender, Canfield and Richmond (2008):
Theorem G S is smooth for any surface S. The proof did not involve an asymptotic counting formula (and indeed none was then known). The method can be adapted to show more. The core of G , core(G ), is the unique maximal subgraph such that the minimum degree δ(G ) ≥ 2. The idea is that, if the core grows reasonably smoothly then rooting trees in it yields a smooth class. The proof method can be adapted to show that any addable minor-closed class is smooth, and indeed more generally. Colin McDiarmid (Oxford)
Graphs from a minor-closed class
21 / 40
Introduction
Properties of graph classes
Counting
More generally
Well-behaved graph classes Our results will involve a well-behaved class of graphs A. We require that A is proper, minor-closed and bridge-addable, and satisfies certain further conditions. The following classes of graphs are all well-behaved: any proper, minor-closed, addable class (for example the class of forests, or series-parallel graphs or planar graphs); the class G S of graphs embeddable on any given surface S; the class of all graphs which contain at most k vertex-disjoint cycles. The definition of well-behaved requires A also to be ‘freely-addable-or-limited’. [It suffices also for A to be closed under subdivision of edges, if there is a growth constant.]
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
22 / 40
Introduction
Properties of graph classes
Counting
More generally
Well-behaved graph classes Freely-addable-or-limited classes of graphs
H ∈ A is freely addable to A if the disjoint union G ∪ H ∈ A whenever G ∈ A. (If A is decomposable then each graph in A is freely addable.) H ∈ A is limited in A if kH is not in A for some positive integer k. If A is G S then the freely addable graphs are the planar graphs, and the limited graphs are the non-planar graphs in G S . A is freely-addable-or-limited if each graph in A is either freely addable or limited (it cannot be both). Decomposable classes, G S and Ex(kC3 ) are all freely-addable-or-limited. Ex(C3 ∪ C4 ) is not freely-addable-or-limited.
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
23 / 40
Introduction
Properties of graph classes
Counting
More generally
Smoothness and core(Rn ) theorem Theorem Let A be well-behaved, with growth constant γ > e; let C denote the class of connected graphs in A; and let Rn ∈u A. Then both A and C are smooth. (a) C δ≥2 has growth constant β where β is the unique root > 1 to βe 1/β = γ; (b) Let α = 1 − x where x is the unique root < 1 to xe −x = 1/β. Then P(|v (core(Rn )) − αn| > n) = e −Ω(n) . (b) Let D denote the class of connected graphs freely addable to A. Let ρ = 1/γ. Then T (ρ) < D(ρ) < ∞, and the probability that core(Rn ) is connected tends to e T (ρ)−D(ρ) . Conjecture: is smooth? Colin McDiarmidevery (Oxford)minor closed Graphsclass from a minor-closed class
24 / 40
Introduction
Properties of graph classes
Counting
More generally
Graphs on surfaces smoothness
Let us illustrate the last theorem for G S . We have known since 2008 that G S has growth constant γ, where γ is the planar graph growth constant; and from Gim´enez and Noy (2009) we have γ ≈ 27.226878. [Counting G S was vastly improved in 2011 by Chapuy, Fusy, Gim´enez, Mohar and Noy, and by Bender and Gao, to give an asymptotic formula for |GnS |.] G S is well-behaved, and so G S is smooth, as we saw earlier – here we learn something new about the core.
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
25 / 40
Introduction
Properties of graph classes
Counting
More generally
Example: graphs on surfaces core(Rn ) for Rn ∈u G S
Solving βe 1/β = γ gives β ≈ 26.207554. This is the growth constant of the class of (connected) graphs in G S with minimum degree at least 2. The growth constant β is only slightly larger than the growth constant ≈ 26.18412 for 2-connected graphs in G S , from Bender, Gao and Wormald (2002). Solving α = 1 − 1/β gives α ≈ 0.961843; and for Rn ∈u G S v (core(Rn )) ∼ αn whp Also, the asymptotic number αn of vertices in the core of Rn is only slightly larger than the number of vertices in the largest block of Rn , which is about 0.95982n, from Gim´enez, Noy and Ru´e (2007). See also recent work of Noy and Ramos (2014). Colin McDiarmid (Oxford)
Graphs from a minor-closed class
26 / 40
Introduction
Properties of graph classes
Counting
More generally
Example: graphs on surfaces Rn ∈u G S , connectivity of core(Rn )
Finally, consider the connectivity of the 2-core. The class D of connected freely addable graphs is the class of connected planar graphs. From Gim´enez and Noy (2009), e −D(ρ) ≈ 0.963253 where ρ = 1/γ. Further e T (ρ) ≈ 1.038138, so by part (d) of the last Theorem, the probability that core(Rn ) is connected ≈ 0.999990. Thus P(core(Rn ) not connected ) ≈ 10−5 . For comparison P(Frag(Rn ) = C3 ) ∼ e −D(ρ) ρ3 /6 ≈ 8 · 10−6 .
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
27 / 40
Introduction
Properties of graph classes
Counting
More generally
Boltzmann Poisson random graph
Let A be decomposable. Fix ρ > 0 such that A(ρ) is finite; and let µ(H) =
ρv (H) for each H ∈ UA. aut(H)
Here UA denotes the unlabelled graphs in A. Easy to see: X A(ρ) = µ(H). H∈U A
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
28 / 40
Introduction
Properties of graph classes
Counting
More generally
Boltzmann Poisson random graph
The Boltzmann Poisson random graph R = R(A, ρ) takes values in UA, with µ(H) P[R = H] = for each H ∈ UA. A(ρ) Let C denote the class of connected graphs in A. For each H ∈ UC let κ(G , H) be the number of components of G isomorphic to H.
Proposition The random variables κ(R, H) for H ∈ UC are independent, with κ(R, H) ∼ Po(µ(H)).
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
29 / 40
Introduction
Properties of graph classes
Counting
More generally
Fragments theorem Recall that H ∈ A is freely addable to A if the disjoint union G ∪ H ∈ A whenever G ∈ A. Let FA be the class of graphs freely addable to A. Observe that FA is decomposable. Also, if A is bridge-addable then so is FA , and then FA is addable.
Theorem Let A be well-behaved, and let ρ = ρ(A). Let FA be the class of graphs freely addable to A, with egf FA . Then 0 < ρ < ∞ and FA (ρ) is finite; and for Rn ∈u A, Fn = UFrag(Rn ) satisfies Fn →d R where R is the Boltzmann Poisson random graph R(FA , ρ).
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
30 / 40
Introduction
Properties of graph classes
Counting
More generally
Corollaries on Fragments and connectivity Corollary Let D be the class of connected graphs in FA . Given distinct graphs H1 , . . . , Hk in UD the k random variables κ(Fn , Hi ) are asymptotically independent with distribution Po(µ(Hi )). This gives for example, for Rn ∈u A P(Rn is connected ) → e −D(ρ) . Consider trees T and forests F, where ρ = 1/e. For Rn ∈u F, since T (ρ) = 12 , 1 |Tn | P(Rn is connected ) = → e− 2 . |Fn | Colin McDiarmid (Oxford)
Graphs from a minor-closed class
31 / 40
Introduction
Properties of graph classes
Counting
More generally
Other minor-closed classes Connected excluded minors
What behaviour can we see with other minor-closed classes, not well-behaved? Example Path forests, ie Ex(C3 , K1,3 ). Decomposable but not bridge-addable. Smooth with growth constant 1. √ κ(Rn ) asymptotically normal, mean ∼ n. √ Largest component size ∼ n log n. More examples in: M. Bousquet-M´elou and K. Weller (2014) Asymptotic properties of some minor-closed classes of graphs Colin McDiarmid (Oxford)
Graphs from a minor-closed class
32 / 40
Introduction
Properties of graph classes
Counting
More generally
Other minor-closed classes Disconnected excluded minors
Example At most k disjoint cycles, ie Ex((k + 1)C3 ). Not decomposable. Looks like a forest with k additional ‘free’ vertices. Smooth with growth constant γk = 2k e. P(Rn ) is connected) → pk := e −T (1/γk )
1
(p0 = e − 2 ).
Similar behaviour for Ex((k + 1)Ct ), Ex((k + 1)D), Ex((k + 1)K1,t ),.. (and for unlabelled graphs with few disjoint cycles) but not for Ex(2K4 ). Kurauskas and McD (2011, 2012), McD (2011), Kang and McD 2011.
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
33 / 40
Introduction
Properties of graph classes
Counting
More generally
Disjoint excluded minors disjoint minors from an addable class
Let apexk A denote the set of G such that there is a set X of at most k vertices with G − X ∈ A. A fan is a path P together with a vertex adjacent to each vertex on P. Let A be addable, with set B of (2-connected) excluded minors (so A = Ex(B)). If A does not contain all fans, then Ex(k + 1)B is the union of apexk A and an exponentially smaller class. If A contains all fans (eg A = Ex(K4 )) then this is false. [Ex(K1,t ) is not addable. Disjoint K1,t minors behave a little differently for t ≥ 4: the difference class is smaller by a factor 2−Θ(n Colin McDiarmid (Oxford)
Graphs from a minor-closed class
2t−5 2t−4 )
. ] 34 / 40
Introduction
Properties of graph classes
Counting
More generally
Disjoint excluded minors disjoint minors from an addable class
Let apexk A denote the set of G such that there is a set X of at most k vertices with G − X ∈ A. A fan is a path P together with a vertex adjacent to each vertex on P. Let A be addable, with set B of (2-connected) excluded minors (so A = Ex(B)). If A does not contain all fans, then Ex(k + 1)B is the union of apexk A and an exponentially smaller class. If A contains all fans (eg A = Ex(K4 )) then this is false. [Ex(K1,t ) is not addable. Disjoint K1,t minors behave a little differently for t ≥ 4: the difference class is smaller by a factor 2−Θ(n Colin McDiarmid (Oxford)
Graphs from a minor-closed class
2t−5 2t−4 )
. ] 34 / 40
Introduction
Properties of graph classes
Counting
More generally
Disjoint K4 minors 1
What about Ex(2K4 )? Ex(K4 ) has growth constant γ ≈ 9.07 (Bernardi, Noy and Welsh 2010). So apex(Ex(K4 )) has growth constant 2γ. Ex(2K4 ) ⊇ apex3 F. apex3 F has growth constant 23 e > 2γ. Thus apex(Ex(K4 )) is exponentially smaller than Ex(2K4 ).
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
35 / 40
Introduction
Properties of graph classes
Counting
More generally
Disjoint K4 minors 1
What about Ex(2K4 )? Ex(K4 ) has growth constant γ ≈ 9.07 (Bernardi, Noy and Welsh 2010). So apex(Ex(K4 )) has growth constant 2γ. Ex(2K4 ) ⊇ apex3 F. apex3 F has growth constant 23 e > 2γ. Thus apex(Ex(K4 )) is exponentially smaller than Ex(2K4 ).
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
35 / 40
Introduction
Properties of graph classes
Counting
More generally
Disjoint K4 minors 1
What about Ex(2K4 )? Ex(K4 ) has growth constant γ ≈ 9.07 (Bernardi, Noy and Welsh 2010). So apex(Ex(K4 )) has growth constant 2γ. Ex(2K4 ) ⊇ apex3 F. apex3 F has growth constant 23 e > 2γ. Thus apex(Ex(K4 )) is exponentially smaller than Ex(2K4 ).
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
35 / 40
Introduction
Properties of graph classes
Counting
More generally
Disjoint K4 minors 2
A model of H in G is a subgraph of G contractible to H. Observe: apexk A is the set of G such that there is a set X of at most k vertices satisfying |X ∩ V (B)| ≥ 1 for each model B of a graph in B. Name it again as 1BLk (B). Let jBLk (B) denote the set of G such that there is a set X of at most k vertices satisfying |X ∩ V (B)| ≥ j for each model B of a graph in B. apex3 F ⊆ 2BL3 K4 ⊆ Ex(2K4 ).
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
36 / 40
Introduction
Properties of graph classes
Counting
More generally
Disjoint K4 minors 3
Kurauskas and McD (2012) conjectured that Ex(2K4 ) is the union of 2BL3 (K4 ) and an exponentially smaller class; and more generally Ex((k + 1)K4 ) is the union of 2BL2k+1 (K4 ) and an exponentially smaller class. This has recently been proved by Valentas Kurauskas (2013).
For each j there are graphs H such that Ex(2H) is the union of jBL2j−1 (H) and an exponentially smaller class, and no smaller j 0 works....
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
37 / 40
Introduction
Properties of graph classes
Counting
More generally
Disjoint K4 minors 3
Kurauskas and McD (2012) conjectured that Ex(2K4 ) is the union of 2BL3 (K4 ) and an exponentially smaller class; and more generally Ex((k + 1)K4 ) is the union of 2BL2k+1 (K4 ) and an exponentially smaller class. This has recently been proved by Valentas Kurauskas (2013).
For each j there are graphs H such that Ex(2H) is the union of jBL2j−1 (H) and an exponentially smaller class, and no smaller j 0 works....
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
37 / 40
Introduction
Properties of graph classes
Counting
More generally
General model Binomial random graph Gn,p
In the classical binomial random graph Gn,p on the vertex set [n], the possible edges appear independently with probability p, 0 < p < 1.
n 2
For each H ∈ An P(Gn,p = H|Gn,p
n λe(H) p e(H) (1 − p)(2)−e(H) P = ∈ A) = P n e(G ) p e(G ) (1 − p)(2)−e(G ) G ∈An λ
G ∈An
where λ = p/(1 − p). Here we assume that An 6= ∅, and e(G ) denotes the number of edges in G .
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
38 / 40
Introduction
Properties of graph classes
Counting
More generally
General model Random cluster model
In the more general random-cluster model, we are also given ν > 0; and the random graph Rn ranges over the graphs H on [n], with n P(Rn = H) ∝ p e(H) (1 − p)(2)−e(H) · ν κ(H) .
Here κ(H) denotes the number of components of H. For each H ∈ An we have λe(H) ν κ(H) P(Rn = H | Rn ∈ A) = P . e(G ) ν κ(G ) G ∈An λ
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
39 / 40
Introduction
Properties of graph classes
Counting
More generally
General model New model
The distribution of our random graphs in A is as follows. Given edge-parameter λ > 0 and component-parameter ν > 0, we let the weighting τ be the pair (λ, ν). For each P graph G we let τ (G ) = λe(G ) ν κ(G ) ; and we denote G ∈An τ (G ) by τ (An ). Rn ∈τ A means that Rn is a random graph which takes values in An with P(Rn = H) =
τ (H) . τ (An )
We call Rn a τ -weighted random graph from A. When λ = ν = 1 we are back to random graphs sampled uniformly. Think of this case! Colin McDiarmid (Oxford)
Graphs from a minor-closed class
40 / 40
Properties of graph classes
Counting
More generally
Random graphs from a minor-closed class Colin McDiarmid Oxford University
Analysis of Algorithms, Paris, June 2014
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
1 / 40
Introduction
Properties of graph classes
Counting
More generally
Question
Let A be a class of (simple) graphs closed under isomorphism, eg the class P of planar graphs. An is the set of graphs in A on vertices 1, . . . , n. Rn ∈u A means that Rn is picked uniformly at random from An . What are typical properties of Rn ? usually a giant component? probability of being connected? many vertices of degree 1? size of the 2-core? Can we learn anything useful for the design or analysis of algorithms?
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
2 / 40
Introduction
Properties of graph classes
Counting
More generally
Generating functions
Given P a class A of graph, A(x) denotes the exponential generating function (egf) n |An |x n /n!. Also ρ = ρ(A) = ρ(A) is the radius of convergence. For suitable classes of graphs, we can relate the egfs (or two variable versions) of all graphs, connected graphs, 2-connected graphs and 3-connected graphs. If we know enough about the 3-connected graphs (as we do for planar graphs, thanks to Tutte and others) then we may be able to extend to all graphs. Let us leave that for now and proceed in greater generality.
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
3 / 40
Introduction
Properties of graph classes
Counting
More generally
Minors H is a minor of G if H can be obtained from a subgraph of G by edge-contractions. A is minor-closed if G ∈ A, H a minor of G
⇒ H∈A
Examples: forests, series-parallel graphs, and more generally graphs of treewidth ≤ k; outerplanar graphs, planar graphs, and more generally graphs embeddable on a given surface; graphs with at most k (vertex) disjoint cycles.
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
4 / 40
Introduction
Properties of graph classes
Counting
More generally
Minors
Ex(H) is the class of graphs with no minor a graph in H. For example: series-parallel graphs = Ex(K4 ), planar graphs = Ex({K5 , K3,3 }), graphs with no two disjoint cycles = Ex(2C3 ). Easy to see that: A is minor-closed iff A = Ex(H) for some class H. Robertson and Seymour’s graph minors theorem (once Wagner’s conjecture) is that if A is minor-closed then A = Ex(H) for some finite class H. The unique minimal such H consists of the excluded minors for A.
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
5 / 40
Introduction
Properties of graph classes
Counting
More generally
Minors Mostly we shall assume that A is minor-closed and proper (that is, not empty and not all graphs). For such A, a result of Mader says: there is a c = c(A) such that the average degree of each graph in A is at most c. Thus our √ graphs are sparse. For Ex(Kt ) the maximum average degree is of order t log t (Kostochka, Thomason). Call A small if ρ(A) > 0, that is ∃c such that |An | ≤ c n n! . Norine, Seymour, Thomas and Wollan (2006); and Dvor´ak and Norine (2010) showed that: Each (proper) minor-closed graph class A is small.
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
6 / 40
Introduction
Properties of graph classes
Counting
More generally
Decomposable If a graph is in A if and only if each component is, then we call A decomposable. For example the class of planar graph is decomposable but the class of graphs embeddable on the torus is not. A minor-closed class is decomposable iff each excluded minor is connected.
Let A be a decomposable class of graphs; and let C consist of the connected graphs in A, with egf C (x). The exponential formula says that A(x) = e C (x) .
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
7 / 40
Introduction
Properties of graph classes
Counting
More generally
Bridge-addable and addable
A is bridge-addable if whenever G ∈ A and u and v are in different components of G then G + uv ∈ A. A is addable if it is decomposable and bridge-addable. A minor-closed class A is addable iff each excluded minor is 2-connected. G S is bridge-addable but not decomposable (and so not addable) except in the planar case.
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
8 / 40
Introduction
Properties of graph classes
Counting
More generally
Bridge-addability and being connected
From McD, Steger and Welsh (2005):
Lemma If A is bridge-addable and Rn ∈u A then P(Rn is connected) ≥ 1/e. 1
For trees T and forests F, |Tn | = nn−2 and |Fn | ∼ e 2 nn−2 . Thus for Fn ∈u F, 1 P(Fn is connected) ∼ e − 2 .
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
9 / 40
Introduction
Properties of graph classes
Counting
More generally
Bridge-addability and being connected
McD, Steger and Welsh (2006) conjectured:
Conjecture 1
If A is bridge-addable then P(Rn is connected) ≥ e − 2 +o(1) . Balister, Bollob´as and Gerke (2008, 2010) give an asymptotic lower bound of e −0.7983 . Norine (2013) improves this to e −2/3 , but the full conjecture is still open. Addario-Berry, McD and Reed (2012), and Kang and Panagiotou (2013) prove the conjecture if A is also closed under deleting bridges, that is if A is bridge-alterable.
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
10 / 40
Introduction
Properties of graph classes
Counting
More generally
Bridge-alterability and connectivity
Here is a natural strengthening of the last conjecture, see eg Balister, Bollob´as and Gerke (2010).
Conjecture Let A be bridge-addable, Rn ∈u A and Fn ∈u F. Then P(Rn is connected) ≥ P(Fn is connected). 1
(Recall that P(Fn is connected) ∼ e − 2 .)
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
11 / 40
Introduction
Properties of graph classes
Counting
More generally
Bridge-alterability and connectivity
The result below (2013) gives a weakened form of the conjecture.
Theorem Let A be bridge-alterable, Rn ∈u A, and Ft ∈u F for t = 1, 2, . . .. Then P(Rn is connected) ≥
min P(Ft is connected).
n/3 0 such that Pr (Rn has < αn pendant copies of H) = e −Ω(n) . Often this shows that there are linear numbers of vertices of each degree, and exponentially many automorphisms. For Rn ∈u P, whp ω(Rn ) = 4 and so χ(Rn ) = 4. Hadwiger’s Conjecture being false says that for some k, there is a graph G ∈ Ex(Kk ) with χ(G ) ≥ k. But then for Rn ∈u Ex(Kk ), wvhp χ(Rn ) ≥ k. All but an exponentially small proportion of (Kk ) are counterexamples! Colin McDiarmid (Oxford)
Graphs from a minor-closed class
18 / 40
Introduction
Properties of graph classes
Counting
More generally
Pendant copies theorem Theorem Let A have a finite positive growth constant, and let H be freely attachable to A. Let Rn ∈u A. Then there exists α > 0 such that Pr (Rn has < αn pendant copies of H) = e −Ω(n) . Often this shows that there are linear numbers of vertices of each degree, and exponentially many automorphisms. For Rn ∈u P, whp ω(Rn ) = 4 and so χ(Rn ) = 4. Hadwiger’s Conjecture being false says that for some k, there is a graph G ∈ Ex(Kk ) with χ(G ) ≥ k. But then for Rn ∈u Ex(Kk ), wvhp χ(Rn ) ≥ k. All but an exponentially small proportion of (Kk ) are counterexamples! Colin McDiarmid (Oxford)
Graphs from a minor-closed class
18 / 40
Introduction
Properties of graph classes
Counting
More generally
Pendant copies theorem searching for a subgraph
Let A have a growth constant, and let the connected graph H be freely attachable. For example, let A be addable and minor-closed and let H ∈ A; or let A be G S and H be planar. Suppose we want to find a copy of H in Rn or verify there is no such subgraph. How quickly can we do so? We see that we can do so in constant expected time; and similarly if we can seek an induced copy of H or a minor H.
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
19 / 40
Introduction
Properties of graph classes
Counting
More generally
Pendant copies theorem searching for a subgraph
Let A have a growth constant, and let the connected graph H be freely attachable. For example, let A be addable and minor-closed and let H ∈ A; or let A be G S and H be planar. Suppose we want to find a copy of H in Rn or verify there is no such subgraph. How quickly can we do so? We see that we can do so in constant expected time; and similarly if we can seek an induced copy of H or a minor H.
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
19 / 40
Introduction
Properties of graph classes
Counting
More generally
Smoothness
Let A be any small class of graphs. Call A smooth if
|An | n|An−1 |
→ a limit as n → ∞.
In this case the limit must be the growth constant γ. All the classes for which we know an asymptotic counting formula are smooth, for example series-parallel graphs, P, G S . Showing smoothness is an important step in proving results about Rn ∈u A.
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
20 / 40
Introduction
Properties of graph classes
Counting
More generally
When is A smooth? Bender, Canfield and Richmond (2008):
Theorem G S is smooth for any surface S. The proof did not involve an asymptotic counting formula (and indeed none was then known). The method can be adapted to show more. The core of G , core(G ), is the unique maximal subgraph such that the minimum degree δ(G ) ≥ 2. The idea is that, if the core grows reasonably smoothly then rooting trees in it yields a smooth class. The proof method can be adapted to show that any addable minor-closed class is smooth, and indeed more generally. Colin McDiarmid (Oxford)
Graphs from a minor-closed class
21 / 40
Introduction
Properties of graph classes
Counting
More generally
Well-behaved graph classes Our results will involve a well-behaved class of graphs A. We require that A is proper, minor-closed and bridge-addable, and satisfies certain further conditions. The following classes of graphs are all well-behaved: any proper, minor-closed, addable class (for example the class of forests, or series-parallel graphs or planar graphs); the class G S of graphs embeddable on any given surface S; the class of all graphs which contain at most k vertex-disjoint cycles. The definition of well-behaved requires A also to be ‘freely-addable-or-limited’. [It suffices also for A to be closed under subdivision of edges, if there is a growth constant.]
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
22 / 40
Introduction
Properties of graph classes
Counting
More generally
Well-behaved graph classes Freely-addable-or-limited classes of graphs
H ∈ A is freely addable to A if the disjoint union G ∪ H ∈ A whenever G ∈ A. (If A is decomposable then each graph in A is freely addable.) H ∈ A is limited in A if kH is not in A for some positive integer k. If A is G S then the freely addable graphs are the planar graphs, and the limited graphs are the non-planar graphs in G S . A is freely-addable-or-limited if each graph in A is either freely addable or limited (it cannot be both). Decomposable classes, G S and Ex(kC3 ) are all freely-addable-or-limited. Ex(C3 ∪ C4 ) is not freely-addable-or-limited.
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
23 / 40
Introduction
Properties of graph classes
Counting
More generally
Smoothness and core(Rn ) theorem Theorem Let A be well-behaved, with growth constant γ > e; let C denote the class of connected graphs in A; and let Rn ∈u A. Then both A and C are smooth. (a) C δ≥2 has growth constant β where β is the unique root > 1 to βe 1/β = γ; (b) Let α = 1 − x where x is the unique root < 1 to xe −x = 1/β. Then P(|v (core(Rn )) − αn| > n) = e −Ω(n) . (b) Let D denote the class of connected graphs freely addable to A. Let ρ = 1/γ. Then T (ρ) < D(ρ) < ∞, and the probability that core(Rn ) is connected tends to e T (ρ)−D(ρ) . Conjecture: is smooth? Colin McDiarmidevery (Oxford)minor closed Graphsclass from a minor-closed class
24 / 40
Introduction
Properties of graph classes
Counting
More generally
Graphs on surfaces smoothness
Let us illustrate the last theorem for G S . We have known since 2008 that G S has growth constant γ, where γ is the planar graph growth constant; and from Gim´enez and Noy (2009) we have γ ≈ 27.226878. [Counting G S was vastly improved in 2011 by Chapuy, Fusy, Gim´enez, Mohar and Noy, and by Bender and Gao, to give an asymptotic formula for |GnS |.] G S is well-behaved, and so G S is smooth, as we saw earlier – here we learn something new about the core.
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
25 / 40
Introduction
Properties of graph classes
Counting
More generally
Example: graphs on surfaces core(Rn ) for Rn ∈u G S
Solving βe 1/β = γ gives β ≈ 26.207554. This is the growth constant of the class of (connected) graphs in G S with minimum degree at least 2. The growth constant β is only slightly larger than the growth constant ≈ 26.18412 for 2-connected graphs in G S , from Bender, Gao and Wormald (2002). Solving α = 1 − 1/β gives α ≈ 0.961843; and for Rn ∈u G S v (core(Rn )) ∼ αn whp Also, the asymptotic number αn of vertices in the core of Rn is only slightly larger than the number of vertices in the largest block of Rn , which is about 0.95982n, from Gim´enez, Noy and Ru´e (2007). See also recent work of Noy and Ramos (2014). Colin McDiarmid (Oxford)
Graphs from a minor-closed class
26 / 40
Introduction
Properties of graph classes
Counting
More generally
Example: graphs on surfaces Rn ∈u G S , connectivity of core(Rn )
Finally, consider the connectivity of the 2-core. The class D of connected freely addable graphs is the class of connected planar graphs. From Gim´enez and Noy (2009), e −D(ρ) ≈ 0.963253 where ρ = 1/γ. Further e T (ρ) ≈ 1.038138, so by part (d) of the last Theorem, the probability that core(Rn ) is connected ≈ 0.999990. Thus P(core(Rn ) not connected ) ≈ 10−5 . For comparison P(Frag(Rn ) = C3 ) ∼ e −D(ρ) ρ3 /6 ≈ 8 · 10−6 .
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
27 / 40
Introduction
Properties of graph classes
Counting
More generally
Boltzmann Poisson random graph
Let A be decomposable. Fix ρ > 0 such that A(ρ) is finite; and let µ(H) =
ρv (H) for each H ∈ UA. aut(H)
Here UA denotes the unlabelled graphs in A. Easy to see: X A(ρ) = µ(H). H∈U A
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
28 / 40
Introduction
Properties of graph classes
Counting
More generally
Boltzmann Poisson random graph
The Boltzmann Poisson random graph R = R(A, ρ) takes values in UA, with µ(H) P[R = H] = for each H ∈ UA. A(ρ) Let C denote the class of connected graphs in A. For each H ∈ UC let κ(G , H) be the number of components of G isomorphic to H.
Proposition The random variables κ(R, H) for H ∈ UC are independent, with κ(R, H) ∼ Po(µ(H)).
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
29 / 40
Introduction
Properties of graph classes
Counting
More generally
Fragments theorem Recall that H ∈ A is freely addable to A if the disjoint union G ∪ H ∈ A whenever G ∈ A. Let FA be the class of graphs freely addable to A. Observe that FA is decomposable. Also, if A is bridge-addable then so is FA , and then FA is addable.
Theorem Let A be well-behaved, and let ρ = ρ(A). Let FA be the class of graphs freely addable to A, with egf FA . Then 0 < ρ < ∞ and FA (ρ) is finite; and for Rn ∈u A, Fn = UFrag(Rn ) satisfies Fn →d R where R is the Boltzmann Poisson random graph R(FA , ρ).
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
30 / 40
Introduction
Properties of graph classes
Counting
More generally
Corollaries on Fragments and connectivity Corollary Let D be the class of connected graphs in FA . Given distinct graphs H1 , . . . , Hk in UD the k random variables κ(Fn , Hi ) are asymptotically independent with distribution Po(µ(Hi )). This gives for example, for Rn ∈u A P(Rn is connected ) → e −D(ρ) . Consider trees T and forests F, where ρ = 1/e. For Rn ∈u F, since T (ρ) = 12 , 1 |Tn | P(Rn is connected ) = → e− 2 . |Fn | Colin McDiarmid (Oxford)
Graphs from a minor-closed class
31 / 40
Introduction
Properties of graph classes
Counting
More generally
Other minor-closed classes Connected excluded minors
What behaviour can we see with other minor-closed classes, not well-behaved? Example Path forests, ie Ex(C3 , K1,3 ). Decomposable but not bridge-addable. Smooth with growth constant 1. √ κ(Rn ) asymptotically normal, mean ∼ n. √ Largest component size ∼ n log n. More examples in: M. Bousquet-M´elou and K. Weller (2014) Asymptotic properties of some minor-closed classes of graphs Colin McDiarmid (Oxford)
Graphs from a minor-closed class
32 / 40
Introduction
Properties of graph classes
Counting
More generally
Other minor-closed classes Disconnected excluded minors
Example At most k disjoint cycles, ie Ex((k + 1)C3 ). Not decomposable. Looks like a forest with k additional ‘free’ vertices. Smooth with growth constant γk = 2k e. P(Rn ) is connected) → pk := e −T (1/γk )
1
(p0 = e − 2 ).
Similar behaviour for Ex((k + 1)Ct ), Ex((k + 1)D), Ex((k + 1)K1,t ),.. (and for unlabelled graphs with few disjoint cycles) but not for Ex(2K4 ). Kurauskas and McD (2011, 2012), McD (2011), Kang and McD 2011.
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
33 / 40
Introduction
Properties of graph classes
Counting
More generally
Disjoint excluded minors disjoint minors from an addable class
Let apexk A denote the set of G such that there is a set X of at most k vertices with G − X ∈ A. A fan is a path P together with a vertex adjacent to each vertex on P. Let A be addable, with set B of (2-connected) excluded minors (so A = Ex(B)). If A does not contain all fans, then Ex(k + 1)B is the union of apexk A and an exponentially smaller class. If A contains all fans (eg A = Ex(K4 )) then this is false. [Ex(K1,t ) is not addable. Disjoint K1,t minors behave a little differently for t ≥ 4: the difference class is smaller by a factor 2−Θ(n Colin McDiarmid (Oxford)
Graphs from a minor-closed class
2t−5 2t−4 )
. ] 34 / 40
Introduction
Properties of graph classes
Counting
More generally
Disjoint excluded minors disjoint minors from an addable class
Let apexk A denote the set of G such that there is a set X of at most k vertices with G − X ∈ A. A fan is a path P together with a vertex adjacent to each vertex on P. Let A be addable, with set B of (2-connected) excluded minors (so A = Ex(B)). If A does not contain all fans, then Ex(k + 1)B is the union of apexk A and an exponentially smaller class. If A contains all fans (eg A = Ex(K4 )) then this is false. [Ex(K1,t ) is not addable. Disjoint K1,t minors behave a little differently for t ≥ 4: the difference class is smaller by a factor 2−Θ(n Colin McDiarmid (Oxford)
Graphs from a minor-closed class
2t−5 2t−4 )
. ] 34 / 40
Introduction
Properties of graph classes
Counting
More generally
Disjoint K4 minors 1
What about Ex(2K4 )? Ex(K4 ) has growth constant γ ≈ 9.07 (Bernardi, Noy and Welsh 2010). So apex(Ex(K4 )) has growth constant 2γ. Ex(2K4 ) ⊇ apex3 F. apex3 F has growth constant 23 e > 2γ. Thus apex(Ex(K4 )) is exponentially smaller than Ex(2K4 ).
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
35 / 40
Introduction
Properties of graph classes
Counting
More generally
Disjoint K4 minors 1
What about Ex(2K4 )? Ex(K4 ) has growth constant γ ≈ 9.07 (Bernardi, Noy and Welsh 2010). So apex(Ex(K4 )) has growth constant 2γ. Ex(2K4 ) ⊇ apex3 F. apex3 F has growth constant 23 e > 2γ. Thus apex(Ex(K4 )) is exponentially smaller than Ex(2K4 ).
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
35 / 40
Introduction
Properties of graph classes
Counting
More generally
Disjoint K4 minors 1
What about Ex(2K4 )? Ex(K4 ) has growth constant γ ≈ 9.07 (Bernardi, Noy and Welsh 2010). So apex(Ex(K4 )) has growth constant 2γ. Ex(2K4 ) ⊇ apex3 F. apex3 F has growth constant 23 e > 2γ. Thus apex(Ex(K4 )) is exponentially smaller than Ex(2K4 ).
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
35 / 40
Introduction
Properties of graph classes
Counting
More generally
Disjoint K4 minors 2
A model of H in G is a subgraph of G contractible to H. Observe: apexk A is the set of G such that there is a set X of at most k vertices satisfying |X ∩ V (B)| ≥ 1 for each model B of a graph in B. Name it again as 1BLk (B). Let jBLk (B) denote the set of G such that there is a set X of at most k vertices satisfying |X ∩ V (B)| ≥ j for each model B of a graph in B. apex3 F ⊆ 2BL3 K4 ⊆ Ex(2K4 ).
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
36 / 40
Introduction
Properties of graph classes
Counting
More generally
Disjoint K4 minors 3
Kurauskas and McD (2012) conjectured that Ex(2K4 ) is the union of 2BL3 (K4 ) and an exponentially smaller class; and more generally Ex((k + 1)K4 ) is the union of 2BL2k+1 (K4 ) and an exponentially smaller class. This has recently been proved by Valentas Kurauskas (2013).
For each j there are graphs H such that Ex(2H) is the union of jBL2j−1 (H) and an exponentially smaller class, and no smaller j 0 works....
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
37 / 40
Introduction
Properties of graph classes
Counting
More generally
Disjoint K4 minors 3
Kurauskas and McD (2012) conjectured that Ex(2K4 ) is the union of 2BL3 (K4 ) and an exponentially smaller class; and more generally Ex((k + 1)K4 ) is the union of 2BL2k+1 (K4 ) and an exponentially smaller class. This has recently been proved by Valentas Kurauskas (2013).
For each j there are graphs H such that Ex(2H) is the union of jBL2j−1 (H) and an exponentially smaller class, and no smaller j 0 works....
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
37 / 40
Introduction
Properties of graph classes
Counting
More generally
General model Binomial random graph Gn,p
In the classical binomial random graph Gn,p on the vertex set [n], the possible edges appear independently with probability p, 0 < p < 1.
n 2
For each H ∈ An P(Gn,p = H|Gn,p
n λe(H) p e(H) (1 − p)(2)−e(H) P = ∈ A) = P n e(G ) p e(G ) (1 − p)(2)−e(G ) G ∈An λ
G ∈An
where λ = p/(1 − p). Here we assume that An 6= ∅, and e(G ) denotes the number of edges in G .
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
38 / 40
Introduction
Properties of graph classes
Counting
More generally
General model Random cluster model
In the more general random-cluster model, we are also given ν > 0; and the random graph Rn ranges over the graphs H on [n], with n P(Rn = H) ∝ p e(H) (1 − p)(2)−e(H) · ν κ(H) .
Here κ(H) denotes the number of components of H. For each H ∈ An we have λe(H) ν κ(H) P(Rn = H | Rn ∈ A) = P . e(G ) ν κ(G ) G ∈An λ
Colin McDiarmid (Oxford)
Graphs from a minor-closed class
39 / 40
Introduction
Properties of graph classes
Counting
More generally
General model New model
The distribution of our random graphs in A is as follows. Given edge-parameter λ > 0 and component-parameter ν > 0, we let the weighting τ be the pair (λ, ν). For each P graph G we let τ (G ) = λe(G ) ν κ(G ) ; and we denote G ∈An τ (G ) by τ (An ). Rn ∈τ A means that Rn is a random graph which takes values in An with P(Rn = H) =
τ (H) . τ (An )
We call Rn a τ -weighted random graph from A. When λ = ν = 1 we are back to random graphs sampled uniformly. Think of this case! Colin McDiarmid (Oxford)
Graphs from a minor-closed class
40 / 40
