The Graph Minor Theorem says that the collection of finite graphs ordered by the minor relation is a well quasi order. This apparently innocent statement hides a monstrous proof: the original result by Robertson and Seymour is about 500 pages and twenty articles, in which a new and deep branch of Graph Theory has been developed. The theorem is famous and full of consequences both on the theoretical side of Mathematics and in applications, e.g., to Computer Science. But there is no concise proof available, although many attempts have been made. In this talk, arising from one such failed attempts, an analysis of the Graph Minor Theorem is presented. Why is it so hard? Assuming to use the by-now standard Nash-Williams's approach to prove it,we will illustrate a number of methods which allow to solve or circumvent some of the difficulties. Finally, we will show that the core of this line of thought lies in a coherence question which is common to many parts of Mathematics: elsewhere it has been solved, although we were unable to adapt those solutions to the present framework. So, there is hope for a short proof of the Graph Minor Theorem but it will not be elementary.

1. The Graph Minor Theorem
a walk on the wild side of graphs
Dr M Benini, Dr R Bonacina
Università degli Studi dell’Insubria
Logic Seminars
Università degli Studi di Verona,
April 3rd, 2017

2. 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 which
dwarfs any other result in graph theory and may doubtless be
counted among the deepest theorems that mathematics has to
offer. . . (This theorem) inconspicuous though it may look at a
first glance, has made a fundamental impact both outside graph
theory and within.
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3. The statement
Definition 2 (Graph)
A graph G = 〈V ,E〉 is composed by a set V of nodes or vertices, and
a 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
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4. 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 identified
by V (G) = A and E(G) = x,y ∈ E(H): x,y ∈ A .
The notion of subgraph defines an embedding on graphs: G ≤S H
says that there is a map η, the embedding, that allows to retrieve an
image of G inside H.
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5. 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) for
every 0 < i < j ≤ n, vi = vj.
Definition 6 (Connected graph)
A graph is connected when there is at least one path between every
pair of nodes.
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6. The statement
Definition 7 (Minor)
G is a minor of H, G ≤M H, if and only if there is an equivalence
relation ∼ on V (H) whose equivalence classes induce connected
subgraphs in H, and G ≤S H/∼, with V (H/∼) = V (H)/∼ and
E(H/∼) = [x]∼ ,[y]∼ : x ∼ y and x,y ∈ E(H) .
For the sake of brevity, an equivalence inducing connected subgraphs
as above, is called a c-equivalence.
Fact 8
Let G be the collection of all the finite graphs.
Then 〈G;≤S〉 and 〈G;≤M〉 are partial orders.
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7. The statement
Definition 9 (Quasi order)
A quasi order A = 〈A;≤〉 is a class A with a binary relation ≤ on A
which is reflexive and transitive. If the relation is also anti-symmetric,
A is a partial order.
Given x,y ∈ A, x ≤ y means that x and y are not related by ≤; x is
equivalent to y, x y, when x ≤ y and y ≤ x; x is incomparable with
y, x y, when x ≤ y and y ≤ x. The notation x < y means x ≤ y and
x y; x ≥ y is the same as y ≤ x; x > y stands for y < x.
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8. The statement
Definition 10 (Descending chain)
Let A = 〈A;≤〉 be a quasi order. Every sequence {xi ∈ A}i∈I, with I an
ordinal, such that xi ≥ xj for every i < j is a descending chain. If a
descending chain {xi }i∈I is such that xi > xj whenever i < j, then it is a
proper descending chain.
A (proper) descending is finite when the indexing ordinal I < ω. If
every proper descending chain in A is finite, then the quasi order is
said to be well founded.
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9. The statement
Definition 11 (Antichain)
Let A = 〈A;≤〉 be a quasi order. Every sequence {xi ∈ A}i∈I, with I an
ordinal, such that xi xj for every i = j is an antichain.
An antichain is finite when the indexing ordinal I < ω. If every
antichain in A is finite, then the quasi order is said to satisfy the finite
antichain property or, simply, to have finite antichains.
Definition 12 (Well quasi order)
A well quasi order is a well founded quasi order having the finite
antichain property.
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10. Nash-Williams’s toolbox
Definition 13 (Bad sequence)
Let A = 〈A;≤〉 be a quasi order. An infinite sequence {xi }i∈ω in A is
bad if and only if xi ≤ 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’ is
used: a bad sequence {xi }i∈ω is minimal with respect to µ and r in A
when for every bad sequence yi i∈ω such that, for some n ∈ ω, xi r yi
when i < n , it holds that µ(yn) <W µ(xn). Here, µ: A → W is a
function from A to some well founded quasi order 〈W ;≤W 〉 and r is a
reflexive binary relation on A.
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11. Nash-Williams’s toolbox
Theorem 14
Let 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 interest
because it has been used to prove the Graph Minor Theorem. But we
will not illustrate it today.
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12. Nash-Williams’s toolbox
Lemma 15 (Dickson)
Assume A and B to be non empty sets. Then A = 〈A;≤A〉 and
B = 〈B;≤B〉 are well quasi orders if and only if A×B = 〈A×B;≤×〉 is a
well quasi order, with the ordering on the Cartesian product defined
by (x1,y1) ≤× (x2,y2) if and only if x1 ≤A x2 and y1 ≤B y2.
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13. 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 and
m 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 to
a similar result.
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14. 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.
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15. Nash-Williams’s toolbox
Lemma 18
Let A = 〈A;≤A〉 be a quasi order which is not a well quasi order, and
let 〈W ;≤〉 be a well founded quasi order. Also, let f : A → W be a
function and r ⊆ A×A a reflexive relation.
Then, there is a bad sequence {xi }i∈ω on A that is minimal with
respect 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) < f (xn).
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16. Nash-Williams’s toolbox
Let B = 〈B;≤B〉 be a quasi order. Let 〈W ;≤〉 be a total well founded
quasi order, and let µ: B → W be a function. Also, let r be a
reflexive relation on B, which is used to possibly identify distinct
elements in B.
Suppose B is not a well quasi order, then there is {Bi }i∈ω bad in B and
minimal with respect to µ and r by Lemma 18.
Let p ∈ ω and let ∆: Bi : i ≥ p → ℘fin(B), the collection of all the
finite 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 19
Let D = 〈 i>p ∆(Bi );≤B〉. Then D is a well quasi order.
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17. Nash-Williams’s toolbox
Summarising,
we want to prove that B = 〈B;≤B〉 is a well quasi order, and we
know it is a quasi order.
Suppose B is not a well quasi order. Then there is a minimal bad
sequence {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 previous
results (Dickson, Higman, Kruskal, and variants) it is usually easy
to deduce that C lies in a well quasi order.
Then, C contains an increasing pair. So, each component of Bn is
less than a component in Bm.
Recombine the pieces, and it follows (!) that Bn ≤ Bm,
contradicting the initial assumption. Q.E.D.
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18. A non-proof of GMT
We want to prove that G = 〈G;≤M〉 is a well quasi order, and we
know it is a quasi order.
Suppose G is not a well quasi order. Then there is a minimal bad
sequence {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 that
there is finite number of components;
each component is a minor of Bi ;
each component has less edges than Bi ;
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 Bn ≤M Bm
Add back the arcs en and em. Thus Bn ≤M BM. Contradiction.
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19. A non-proof of GMT
The red part is wrong, already on forests:
M
but
=
When adding back edges, they are linked to the ‘wrong’ nodes,
preventing to lift the embedding on the parts to the whole.
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20. Embeddings versus minors
G ≤M H means that there is an embedding of G into H: a map f
from the nodes of G to a quotient on the nodes of H by means of a
c-equivalence such that f preserves arcs.
The problem in the previous non-proof is that the cancelled edges eG
and eH in the decomposition process lie between two nodes which
may be mapped by f in such a way that f (eG) = eH.
But, if one is able to constrain the way nodes are mapped. . .
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21. 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 the
pattern of the proof.
But marking the root and using a minor relation ≤R which preserves
roots, is the key to a concise and elegant proof.
In fact, it suffices to show that the restricted minor relation ≤R is
contained in the usual ≤M to derive the theorem: in fact, if there is a
bad sequence {Bi }i∈ω with respect to ≤M, then it is bad with respect
to ≤R. So, if 〈T ;≤R〉 is a well quasi order, then it cannot contain a
bad sequence, and so neither can 〈T ;≤M〉.
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22. Coherence
The Kruskal Theorem is very nice subcase of the Graph Minor
Theorem, 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 a
sub-relation of ≤M which preserves roots. And this is enough to prove
Kruskal’s Theorem.
It is tempting to extend the idea behind Kruskal’s Theorem to general
graphs. And it is easy: mark the endpoints of the arcs we are going to
delete. Formally, this amounts to prove the Graph Minor Theorem on
labelled graphs, with a few additional hypotheses. And it suffices to
prove this extended theorem for labellings over the well quasi order 2.
The proof develops smoothly as before. . . and, frustratingly, it fails in
the same point.
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23. Coherence
We have tried many variations on the theme, and we learnt how to
move the problem in different parts of the proof. But, still, we have
not found how to solve it.
The nature of the problem is subtle: it is about the coherence of
embeddings. It is easier to explain it by means of an example:
M
=
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24. Coherence
The blue arrows denote the embedding as obtained from the
decomposition step. The red arrows denote the embedding of the
cancelled arc on the left to the cancelled arc on the right. The green
edge is the embedding of the red arc on the left as a result of the blue
embedding.
The blue and the red embedding are not coherent: they to not
preserve the endpoints of arcs.
=
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25. 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 that
the above diagram commutes. The f arrows are suitable embeddings
of the components G of G into the components H of H. When this
happens, we say that the f ’s are coherent, since they can be derived
by factoring F through the inclusions of components.
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26. Coherence
G
G
G
... H
H
H
...
f
f
F
In a way, this diagram suggests to construct a colimit in a suitable
category, the one of graphs and coherent embeddings. It is exactly
the way to synthesise the proof of Kruskal’s Theorem: look for a
relation which forces G to be the minimal object containing all the
components, and such that the considered embeddings can be
combined together, so to construct the F above, which acts as the
co-universal arrow.
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27. Hopes for the future
The illustrated approach almost yields a proof of the Graph Minor
Theorem, except for solving the coherence problem.
It is worth remarking that this problem is not about graph, but about
embeddings: we need a way to identify which embeddings can be
combined so to form a co-universal arrow as in the preceding diagram,
and we need to show that the minor relation arising from them is
contained 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 Minor
Theorem: the coherence problem arises in many branches of
Mathematics, completely unrelated to Graph Theory. And many
solutions have been found. As soon as one identifies that a theorem
in the X theory provides a solution to an instance of the abstract
coherence problem, there are chances it could be reused to write
Q.E.D. after the proof we have illustrated before.
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28. 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 a
finite 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 = y .
Clearly g and g−1 are each other inverses. Thus, a graph can be seen
as a finite, reflexive and symmetric relation.
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29. 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 monomorphism
g(G) → g(H).
Suppose ∼ to be a c-equivalence on H, then
∼ is an equivalence relation and, for every x ∈ V (H), the subgraph
of H induced by [x]∼ is connected.
∼ is an equivalence relation and, for every x in the domain
dom(g(H)) of the relation g(H), every pair of elements in [x]∼ lie
in g(H)∗
, the transitive closure of g(H).
∼ is an equivalence on dom(g(H)) such that ∼ ⊆ g(H)∗
.
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30. A minor surprise
Hence, G ≤M H if and only if there is an equivalence relation ∼ on
dom(g(H)) such that ∼ ⊆ g(H)∗
, and there is
η: 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!
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31. The end
Questions?
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