Logic Seminars JAIST, 6th June 2017

1. The Graph Minor Theorem
a walk on the wild side of graphs
M Benini, R Bonacina
Università degli Studi dell’Insubria
Logic Seminars
JAIST,
June 6th, 2017

2. Graph Minor Theorem
Theorem 1 (Graph Minor)
Let G be the collection of all the finite graphs and let ≤M be the
graph minor relation.
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. A standard proof
With no doubt, the proof of the Graph Minor Theorem is
non-standard: it does not follow a canonical proof pattern, but it
seems an ad-hoc attempt to solve a very difficult problem.
A short and manageable proof of this result would be very welcome.
In fact, the first problem is the size of the proof, since just reading
five hundred pages about a single proof is a major task. But the
second problem is more relevant: even a long proof which follows a
canonical pattern would be manageable, apt to variants, and possible
to translate in different contexts.
Is there a canonical proof of the graph minor theorem?
Why does the standard technique due to Nash-Williams fail?
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4. A standard proof
With these objectives in mind, we tried to prove the Graph Minor
Theorem using Nash-Williams’s technique. And we failed.
But we learnt what is the core of the problem, and this seminar wants
to illustrate the (negative) knowledge we gathered in our attempt.
Also, our attempt shows that the problem is difficult, which was
known in advance, but not impossible: there are still unexplored ways,
and the quest is open.
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5. Nash-Williams’s toolbox
Definition 2 (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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6. Nash-Williams’s toolbox
Theorem 3
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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7. Nash-Williams’s toolbox
Lemma 4 (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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8. 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 5 (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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9. Nash-Williams’s toolbox
Theorem 6 (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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10. Nash-Williams’s toolbox
Lemma 7
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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11. 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 7.
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 8
Let D = 〈 i>p ∆(Bi );≤B〉. Then D is a well quasi order.
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12. 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, for some n < m.
Recombine the pieces, and it follows (!) that Bn ≤ Bm,
contradicting the initial assumption. Q.E.D.
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13. 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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14. 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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15. 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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16. Back to Kruskal
In the case of Kruskal’s Theorem 6, 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 proof. And, applying some care, the
proof explains why ≤M is the ‘right’ relation to look at.
But I already spoke about this in my previous seminar. . .
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17. 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 the way nodes are linked together.
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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18. 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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19. 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 do not
preserve the endpoints of arcs.
=
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20. 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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21. 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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22. Coherence
G
G
... G H
H
... Hf
f
F
In other words, we want to consider the slice categories G/G and
G/H and the morphisms between them, which are the arrows f in the
diagram making the squares to commute.
However, Nash-Williams’s argument does not suffice alone to force
the f embeddings, coming from G ≤M H , to lie in this more
restrictive category. It goes very close to, but not enough yet. . .
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23. 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 generate a morphisms between slice categories. 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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24. 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
equivalently seen as a finite, reflexive and symmetric relation.
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25. 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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26. 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 9 (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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27. The end
Questions?
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