The Graph Minor Theorema walk on the wild side of graphs

Dr M Benini, Dr R Bonacina

Università degli Studi dell’Insubria

Logic SeminarsUniversità degli Studi di Verona,

April 3rd, 2017

Graph Minor Theorem

Theorem 1 (Graph Minor)Let G be the collection of all the finite graphs.Then G= ⟨G ;≤M⟩ is a well quasi order.

Proof.By Robertson and Seymour. About 500 pages, 20 articles.

Quoting Diestel, Graph Theory, 5th ed., Springer (2016):Our goal in this last chapter is a single theorem, one whichdwarfs any other result in graph theory and may doubtless becounted among the deepest theorems that mathematics has tooffer. . . (This theorem) inconspicuous though it may look at afirst glance, has made a fundamental impact both outside graphtheory and within.

( 2 of 31 )

The statement

Definition 2 (Graph)A graph G = ⟨V ,E ⟩ is composed by a set V of nodes or vertices, anda set E of edges or arcs, which are unordered pairs of distinct nodes.Given a graph G , V (G) denotes the set of its nodes and E (G)denotes the set of its edges. A graph G is finite when V (G) is so.

■ No loops■ The definition induces a criterion for equality■ Obvious notion of isomorphism

( 3 of 31 )

The statement

Definition 3 (Subgraph)G is a subgraph of H, G ≤S H, if and only if there is η : V (G)→V (H)injective such that, for every

{x ,y

} ∈E (G),{η(x),η(y)

} ∈E (H).

Definition 4 (Induced subgraph)Let A⊆V (H). Then the induced subgraph G of H by A is identifiedby V (G)=A and E (G)= {{

x ,y} ∈E (H): x ,y ∈A}

.

The notion of subgraph defines an embedding on graphs: G ≤S Hsays that there is a map η, the embedding, that allows to retrieve animage of G inside H.

( 4 of 31 )

The statement

Definition 5 (Path)Let G be a graph and let x ,y ∈V (G). A path p from x to y ,p : x y , of length n ∈N is a sequence

{vi ∈V (G)

}0≤i≤n such that

(i) v0 = x , vn = y , (ii) for every 0≤ i < n, {vi ,vi+1} ∈E (G), and (iii) forevery 0< i < j ≤ n, vi 6= vj .

Definition 6 (Connected graph)A graph is connected when there is at least one path between everypair of nodes.

( 5 of 31 )

The statement

Definition 7 (Minor)G is a minor of H, G ≤M H, if and only if there is an equivalencerelation ∼ on V (H) whose equivalence classes induce connectedsubgraphs in H, and G ≤S H/∼, with V (H/∼)=V (H)/∼ andE (H/∼)= {{

[x ]∼ , [y ]∼}

: x 6∼ y and{x ,y

} ∈E (H)}.

For the sake of brevity, an equivalence inducing connected subgraphsas above, is called a c-equivalence.

Fact 8Let G be the collection of all the finite graphs.Then ⟨G ;≤S⟩ and ⟨G ;≤M⟩ are partial orders.

( 6 of 31 )

The statement

Definition 9 (Quasi order)A quasi order A= ⟨A;≤⟩ is a class A with a binary relation ≤ on Awhich is reflexive and transitive. If the relation is also anti-symmetric,A is a partial order.Given x ,y ∈A, x 6≤ y means that x and y are not related by ≤; x isequivalent to y , x ' y , when x ≤ y and y ≤ x ; x is incomparable withy , x ∥ y , when x 6≤ y and y 6≤ x . The notation x < y means x ≤ y andx 6' y ; x ≥ y is the same as y ≤ x ; x > y stands for y < x .

( 7 of 31 )

The statement

Definition 10 (Descending chain)Let A= ⟨A;≤⟩ be a quasi order. Every sequence {xi ∈A}i∈I , with I anordinal, such that xi ≥ xj for every i < j is a descending chain. If adescending chain {xi }i∈I is such that xi > xj whenever i < j , then it is aproper descending chain.A (proper) descending is finite when the indexing ordinal I <ω. Ifevery proper descending chain in A is finite, then the quasi order issaid to be well founded.

( 8 of 31 )

The statement

Definition 11 (Antichain)Let A= ⟨A;≤⟩ be a quasi order. Every sequence {xi ∈A}i∈I , with I anordinal, such that xi ∥ xj for every i 6= j is an antichain.An antichain is finite when the indexing ordinal I <ω. If everyantichain in A is finite, then the quasi order is said to satisfy the finiteantichain property or, simply, to have finite antichains.

Definition 12 (Well quasi order)A well quasi order is a well founded quasi order having the finiteantichain property.

( 9 of 31 )

Nash-Williams’s toolbox

Definition 13 (Bad sequence)Let A= ⟨A;≤⟩ be a quasi order. An infinite sequence {xi }i∈ω in A isbad if and only if xi 6≤ xj whenever i < j .A bad sequence {xi }i∈ω is minimal in A when there is no bad sequence{yi

}i∈ω such that, for some n ∈ω, xi = yi when i < n and yn < xn.

In fact, in the following, a generalised notion of ‘being minimal’ isused: a bad sequence {xi }i∈ω is minimal with respect to µ and r in Awhen for every bad sequence

{yi

}i∈ω such that, for some n ∈ω, xi r yi

when i < n , it holds that µ(yn) 6<W µ(xn). Here, µ : A→W is afunction from A to some well founded quasi order ⟨W ;≤W ⟩ and r is areflexive binary relation on A.

( 10 of 31 )

Nash-Williams’s toolbox

Theorem 14Let A= ⟨A;≤⟩ be a quasi order. Then, the following are equivalent:1. A is a well quasi order;2. in every infinite sequence {xi }i∈ω in A there exists an increasing

pair xi ≤ xj for some i < j ;

3. every sequence {xi ∈A}i∈ω contains an increasing subsequence{xnj

}j∈ω such that xni ≤ xnj for every i < j .

4. A does not contain any bad sequence.

There is also a finite basis characterisation which is of interestbecause it has been used to prove the Graph Minor Theorem. But wewill not illustrate it today.

( 11 of 31 )

Nash-Williams’s toolbox

Lemma 15 (Dickson)Assume A and B to be non empty sets. Then A= ⟨A;≤A⟩ andB= ⟨B;≤B⟩ are well quasi orders if and only if A×B= ⟨A×B;≤×⟩ is awell quasi order, with the ordering on the Cartesian product definedby (x1,y1)≤× (x2,y2) if and only if x1 ≤A x2 and y1 ≤B y2.

( 12 of 31 )

Nash-Williams’s toolbox

Let A= ⟨A;≤A⟩ be a well quasi order, and define

A∗ = {{xi }i<n : n ∈ω and, for all i < n,xi ∈A

}as the collection of all the finite sequences over A.Then A∗ = ⟨A∗;≤∗⟩ is defined as {xi }i<n ≤∗

{yi

}i<m if and only if there

is η : n→m injective and monotone between the finite ordinals n andm such that xi ≤A yη(i) for all i < n.

Lemma 16 (Higman)A∗ = ⟨A∗;≤∗⟩ is a well quasi order.

Dropping the requirement that η above has to be monotone, leads toa similar result.

( 13 of 31 )

Nash-Williams’s toolbox

Theorem 17 (Kruskal)Let T be the collection of all the finite trees.Then T= ⟨T ;≤M⟩ is a well quasi order.

We remind that a tree is a connected and acyclic graph.Kruskal’s Theorem has much more to reveal than its statement says:for example, it is unprovable in Peano arithmetic.

( 14 of 31 )

Nash-Williams’s toolbox

Lemma 18Let A= ⟨A;≤A⟩ be a quasi order which is not a well quasi order, andlet ⟨W ;≤⟩ be a well founded quasi order. Also, let f : A→W be afunction and r ⊆A×A a reflexive relation.Then, there is a bad sequence {xi }i∈ω on A that is minimal withrespect to f and r : for every n ∈ω and for every bad sequence

{yi

}i∈ω

on A such that xi r yi whenever i < n, f (yn) 6< f (xn).

( 15 of 31 )

Nash-Williams’s toolbox

Let B= ⟨B;≤B⟩ be a quasi order. Let ⟨W ;≤⟩ be a total well foundedquasi order, and let µ : B →W be a function. Also, let r be areflexive relation on B, which is used to possibly identify distinctelements in B.Suppose B is not a well quasi order, then there is {Bi }i∈ω bad in B andminimal with respect to µ and r by Lemma 18.Let p ∈ω and let ∆ :

{Bi : i ≥ p

}→℘fin(B), the collection of all thefinite subsets of B, be such that

(∆1) for every i ∈ω and for every x ∈∆(Bi), x ≤B Bi ;(∆2) for every i ∈ω and for every x ∈∆(Bi), µ(x)<µ(Bi).

Proposition 19Let D= ⟨⋃i>p∆(Bi);≤B⟩. Then D is a well quasi order.

( 16 of 31 )

Nash-Williams’s toolbox

Summarising,■ we want to prove that B= ⟨B;≤B⟩ is a well quasi order, and weknow it is a quasi order.

■ Suppose B is not a well quasi order. Then there is a minimal badsequence {Bi }i∈ω with respect to some reasonable measure µ and =.

■ Define a decomposition ∆ of the elements in the bad sequence.■ Then, the collection of the components forms a well quasi order.■ Form a sequence C from the components: by using the previousresults (Dickson, Higman, Kruskal, and variants) it is usually easyto deduce that C lies in a well quasi order.

■ Then, C contains an increasing pair. So, each component of Bn isless than a component in Bm.

■ Recombine the pieces, and it follows (!) that Bn ≤Bm,contradicting the initial assumption. Q.E.D.

( 17 of 31 )

A non-proof of GMT

■ We want to prove that G= ⟨G ;≤M⟩ is a well quasi order, and weknow it is a quasi order.

■ Suppose G is not a well quasi order. Then there is a minimal badsequence {Bi }i∈ω with respect to

∣∣E (_)∣∣ and =.

■ Fix a sequence{ei ∈E (Bi)

}i>p and define

∆(Bi)={G ′ : V (G ′)=V (G) and E (G ′)=E (G) \ {ei }

}.

It is easy to check thatä there is finite number of components;ä each component is a minor of Bi ;ä each component has less edges than Bi ;ä for some p ∈ω, each Bi , i > p, contains an arc.

■ Then, the collection of the components D forms a well quasi order.■ Construct the sequence C = {

∆(Bi)}i>p: it lies in D.

■ Then, C contains an increasing pair B′n ≤M B′

m■ Add back the arcs en and em. Thus Bn ≤M BM . Contradiction.

( 18 of 31 )

A non-proof of GMT

The red part is wrong, already on forests:

6∥M

but

=

When adding back edges, they are linked to the ‘wrong’ nodes,preventing to lift the embedding on the parts to the whole.

( 19 of 31 )

Embeddings versus minors

G ≤M H means that there is an embedding of G into H: a map ffrom the nodes of G to a quotient on the nodes of H by means of ac-equivalence such that f preserves arcs.The problem in the previous non-proof is that the cancelled edges eGand eH in the decomposition process lie between two nodes whichmay be mapped by f in such a way that f (eG) 6= eH .But, if one is able to constrain the way nodes are mapped. . .

( 20 of 31 )

Back to Kruskal

In the case of Kruskal’s Theorem 17, the problem is easy to solve:deleting an arc yields two disjoint subtrees.Considering them as two distinct components does not change thepattern of the proof.But marking the root and using a minor relation ≤R which preservesroots, is the key to a concise and elegant proof.In fact, it suffices to show that the restricted minor relation ≤R iscontained in the usual ≤M to derive the theorem: in fact, if there is abad sequence {Bi }i∈ω with respect to ≤M , then it is bad with respectto ≤R . So, if ⟨T ;≤R⟩ is a well quasi order, then it cannot contain abad sequence, and so neither can ⟨T ;≤M⟩.

( 21 of 31 )

Coherence

The Kruskal Theorem is very nice subcase of the Graph MinorTheorem, which points to the core of the problem.It is not the independence of the components, like in Kruskal’s case,that marks the difference. Rather, it is the fact that we can identify asub-relation of ≤M which preserves roots. And this is enough to proveKruskal’s Theorem.It is tempting to extend the idea behind Kruskal’s Theorem to generalgraphs. And it is easy: mark the endpoints of the arcs we are going todelete. Formally, this amounts to prove the Graph Minor Theorem onlabelled graphs, with a few additional hypotheses. And it suffices toprove this extended theorem for labellings over the well quasi order 2.The proof develops smoothly as before. . . and, frustratingly, it fails inthe same point.

( 22 of 31 )

Coherence

We have tried many variations on the theme, and we learnt how tomove the problem in different parts of the proof. But, still, we havenot found how to solve it.The nature of the problem is subtle: it is about the coherence ofembeddings. It is easier to explain it by means of an example:

6∥M

=

( 23 of 31 )

Coherence

The blue arrows denote the embedding as obtained from thedecomposition step. The red arrows denote the embedding of thecancelled arc on the left to the cancelled arc on the right. The greenedge is the embedding of the red arc on the left as a result of the blueembedding.The blue and the red embedding are not coherent: they to notpreserve the endpoints of arcs.

=

( 24 of 31 )

Coherence

Coherence can be described in an abstract way:

G

G ′

G ′

... H

H ′

H ′

...

f

f

F

Coherence amounts to require that there is an embedding F such thatthe above diagram commutes. The f arrows are suitable embeddingsof the components G ′ of G into the components H ′ of H. When thishappens, we say that the f ’s are coherent, since they can be derivedby factoring F through the inclusions of components.

( 25 of 31 )

Coherence

G

G ′

G ′

... H

H ′

H ′

...

f

f

F

In a way, this diagram suggests to construct a colimit in a suitablecategory, the one of graphs and coherent embeddings. It is exactlythe way to synthesise the proof of Kruskal’s Theorem: look for arelation which forces G to be the minimal object containing all thecomponents, and such that the considered embeddings can becombined together, so to construct the F above, which acts as theco-universal arrow.

( 26 of 31 )

Hopes for the future

The illustrated approach almost yields a proof of the Graph MinorTheorem, except for solving the coherence problem.It is worth remarking that this problem is not about graph, but aboutembeddings: we need a way to identify which embeddings can becombined so to form a co-universal arrow as in the preceding diagram,and we need to show that the minor relation arising from them iscontained in ≤M . Then, as in the Kruskal’s case, the result follows.But this fact gives hopes to find a concise proof of the Graph MinorTheorem: the coherence problem arises in many branches ofMathematics, completely unrelated to Graph Theory. And manysolutions have been found. As soon as one identifies that a theoremin the X theory provides a solution to an instance of the abstractcoherence problem, there are chances it could be reused to writeQ.E.D. after the proof we have illustrated before.

( 27 of 31 )

A minor surprise

Let G be a finite graph: define

g(G)= {(x ,y) :

{x ,y

} ∈E (G)}∪{

(x ,x) : x ∈V (G)}

.

Then g(G) is a finite reflexive and symmetric relation over a finite set.Conversely, if γ is a finite reflexive and symmetric relation over afinite set, define g−1(γ) as the graph G such that V (G)= {

x : x γ x}

and E (G)= {{x ,y

}: x γ y and x 6= y

}.

Clearly g and g−1 are each other inverses. Thus, a graph can be seenas a finite, reflexive and symmetric relation.

( 28 of 31 )

A minor surprise

If G ≤S H then■ there is η : V (G)→V (H) injective such that, for every{

x ,y} ∈E (G),

{η(x),η(y)

} ∈E (H).■ there is η injective such that, whenever (x ,y) ∈ g(G),

(η(x),η(y)) ∈ g(H), that is, there is a pointwise monomorphismg(G)→ g(H).

Suppose ∼ to be a c-equivalence on H, then■ ∼ is an equivalence relation and, for every x ∈V (H), the subgraphof H induced by [x ]∼ is connected.

■ ∼ is an equivalence relation and, for every x in the domaindom(g(H)) of the relation g(H), every pair of elements in [x ]∼ liein g(H)∗, the transitive closure of g(H).

■ ∼ is an equivalence on dom(g(H)) such that ∼⊆ g(H)∗.

( 29 of 31 )

A minor surprise

Hence, G ≤M H if and only if there is an equivalence relation ∼ ondom(g(H)) such that ∼⊆ g(H)∗, and there isη : dom(g(G))→ dom(g(H))/∼ injective such that{(η(x),η(y)) : (x ,y) ∈ g(G)

}= η(g(G))⊆ g(H)/∼={([x ]∼, [y ]∼) : (x ,y) ∈ g(H)

}.

Let γ and δ be finite, reflexive and symmetric relations. Define γ≤R δ

if and only if there is an equivalence ∼ on dom(δ) such that ∼⊆ δ∗,and there is η : dom(γ)→ dom(δ)/∼ injective such that η(γ)⊆ δ/∼.

Theorem 20 (Graph Minor)Let R be the collection of finite, reflexive and symmetric relations.Then R= ⟨R;≤R⟩ is a well quasi order.So, surprise, the Graph Minor Theorem is not about graphs!

( 30 of 31 )

The end

Questions?

( 31 of 31 )