The famous Kruskal's tree theorem states that the collection of finite trees labelled over a well quasi order and ordered by homeomorphic embedding, forms a well quasi order. Its intended mathematical meaning is that the collection of finite, connected and acyclic graphs labelled over a well quasi order is a well quasi order when it is ordered by the graph minor relation. Oppositely, the standard proof(s) shows the property to hold for trees in the Computer Science's sense together with an ad-hoc, inductive notion of embedding. The mathematical result follows as a consequence in a somewhat unsatisfactory way. In this talk, a variant of the standard proof will be illustrated explaining how the Computer Science and the graph-theoretical statements are strictly coupled, thus explaining why the double statement is justified and necessary.

1. Explaining the Kruskal’s Tree Theorem
Dr M Benini, Dr R Bonacina
Università degli Studi dell’Insubria
Logic Seminars
JAIST,
May 12th, 2017

2. The theorem
Theorem 1 (Kruskal)
The collection T (A) of all the finite trees labelled over a well quasi
order A ordered by homeomorphic embedding, is a well quasi order
T(A) = 〈T (A);≤E 〉.
The T1 ≤E T2 relation means that there is an embedding η of T1 into
T2, i.e., a function which maps the nodes and the arcs of T1 into
those of T2 and preserves the structure of the T1 tree.
It is worth noticing that the statement is not precise, since the
definitions of ‘tree’ and ‘preserving the structure’ are left implicit.
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3. The theorem
In fact, there is an ambiguous point in the statement: it is usually
intended as speaking of trees as some special graphs, with the notion
of embedding captured via the graph minor relation, while it is proved
by using an inductive definition of trees, close to the usual one in
Computer Science, with a natural but ad-hoc notion of embedding.
So, we are speaking of two distinct theorems, and about their
relation. Also, we have many proofs of one of them, while the other is
usually relegated to a footnote in the end, or a quick hint, or an
exercise, when mentioned. But it is the unproved one which is really
used in Mathematics.
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4. The theorem
Definition 2 (Tree)
Let A = 〈A;≤A〉 be a quasi order. A tree T is inductively defined as
1. a single node, called its root;
2. given the trees T1,...,Tn, then a tree is the structure composed by
a node, called the root and T1,...,Tn, called the immediate
subtrees of the root.
A labelled tree (T,l) over A is a tree T and a function l from its
nodes to A.
Definition 3 (Tree)
A tree is a finite, acyclic and connected graph. A labelled tree (T,l)
over the quasi order A = 〈A;≤A〉 is a labelled graph which is a tree.
The two definitions are evidently different: to distinguish them, we
refer to the trees as for the former one as pointed trees.
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5. The theorem
The notion of embedding for the first definition of tree is
Definition 4 (Pointed minor)
Let A = 〈A;≤A〉 be a well quasi order. Let (T,lT ) and (T ,lT ) be
pointed trees. Then (T,lT ) ≤K (T ,lT ) if and only if one of the
following conditions applies
1. there is an immediate subtree (S ,lT ) of (T ,lT ) such that
(T,lT ) ≤K (S ,lT );
2. calling rT and rT the roots of T and T respectively,
lT (rT ) ≤A lT (rT ) and there is an injective map from the
immediate subtrees (S,lT ) of (T,lT ) to those (S ,lT ) of (T ,lT )
such that (S,lT ) ≤K (S ,lT ).
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6. Well quasi orders
Definition 5 (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.
Intuitively, a quasi order is an order in which we admit two elements
to be equivalent but not equal.
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7. Well quasi orders
Definition 6 (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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8. Well quasi orders
Definition 7 (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 8 (Well quasi order)
A well quasi order is a well founded quasi order having the finite
antichain property.
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9. Nash-Williams’s toolbox
Definition 9 (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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10. Nash-Williams’s toolbox
Theorem 10 (Characterisation)
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.
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11. Nash-Williams’s toolbox
Fact 11
Let A = 〈A;≤〉 be a well quasi order. Then, for every quasi order
A+
= A;≤+
such that ≤ ⊆ ≤+
, A+
is a well quasi order.
Fact 12
Let A = 〈A;≤A〉 be a well quasi order. Then, every quasi order
B = 〈B;≤B〉 with B ⊆ A and ≤B the restriction of ≤A to B, is a well
quasi order.
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12. Nash-Williams’s toolbox
Proposition 13
Let 〈A;≤〉 be a well quasi order and let ≈ be an equivalence relation
on A such that ≤≈ is a quasi ordering of A/≈, with [x]≈ ≤≈ [y]≈ if and
only if there are x ∈ [x]≈ and y ∈ [y]≈ such that x ≤ y . Then
A/≈;≤≈ is a well quasi order.
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13. Nash-Williams’s toolbox
Lemma 14 (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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14. Nash-Williams’s toolbox
Lemma 15
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).
So, if A is a quasi order, but not a well quasi order, then it contains a
bad sequence which is minimal with respect to some measure f and
some comparison criterion r, normally =.
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15. 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.
Suppose B is not a well quasi order, then there is {Bi }i∈ω bad in B and
minimal with respect to µ and = by Lemma 15.
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 16
Let D = 〈 i>p ∆(Bi );≤B〉. Then D is a well quasi order.
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16. 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 well known
results, e.g., Dickson’s Lemma, 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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17. The proof
Theorem 17 (Kruskal)
Let R (A) be the collection of pointed trees over A = 〈A;≤A〉. If A is
a well quasi order then R(A) = R (A);≤K is a well quasi order.
Proof. (i)
Suppose R(A) is not a well quasi order. Then, by Lemma 15 there is
a bad sequence (Ti ,li ) i∈ω in R(A) minimising |E (_)|.
Let (Ti ,li ) i∈I be the subsequence of (Ti ,li ) i∈ω composed by the
pointed trees with no edges. Then they contain just a single node,
the root ri , so li (ri ) i∈I is a sequence in A with no increasing pair.
Thus, by Theorem 10 on the A well quasi order, I is finite, so
p = maxI is defined and (Ti ,li ) i>p is such that E (Ti ) > 0 and, in
particular, there is an edge from the root to some node. →
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18. The proof
→ Proof. (ii)
For i > p, define ∆(Ti ,li ) as the set composed by the two connected
components T1
i ,li , T2
i ,li obtained deleting some arc
{ri ,xi } ∈ E (Ti ): each component is a pointed tree having one endpoint
of {ri ,xi } as its root. We stipulate that the root of T1
i is ri and the
root of T2
i is xi . Clearly, if Tj
i ,li ∈ ∆(Ti ,li ), Tj
i ,li ≤K (Ti ,li ) and
E Tj
i ,li < E (Ti ,li ) . So D = i>p ∆(Ti ,li );≤K is a well quasi
order by Lemma 16. Thus, by Dickson’s Lemma 14, D×D is a well
quasi order. Considering the sequence T1
i ,li , T2
i ,li i>p
, by
Theorem 10 there are m > n such that T1
n ,ln ≤K T1
m,lm , thus
ln (rn) ≤A lm (rm), and T2
n ,ln ≤K T2
m,lm , and the endpoints of the
arc deleted by ∆ are similarly preserved, so (Tn,ln) ≤K (Tm,lm),
contradicting (Ti ,li ) i∈ω to be bad.
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19. Pointed trees versus graphs
Consider the following pair of incomparable trees, and decompose
them as in the previous proof:
but =
The decomposition yields two pairs of subtrees which are identical as
graphs but different as pointed trees.
Thus, extending the proof of Kruskal’s Theorem to trees seen as
graphs is not immediate.
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20. Trees as graphs
Definition 18 (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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21. Trees as graphs
Definition 19 (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 20 (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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22. Trees as graphs
Definition 21 (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 22 (Connected graph)
A graph is connected when there is at least one path between every
pair of nodes.
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23. Trees as graphs
Definition 23 (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 24
Let G be the collection of all the finite graphs.
Then 〈G;≤S〉 and 〈G;≤M〉 are partial orders.
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24. The proof, part II
Theorem 25 (Kruskal)
Let T (A) be the collection of all the pointed trees labelled over
A = 〈A;≤A〉. If A is a well quasi order, then T(A) = T (A);≤A
M is a
well quasi order.
Proof. (i)
Notice how (T,lT ) ≤K (T ,lT ) implies (T,lT ) ≤A
M
(T ,lT ). In fact, a
simple induction on Definition 4 suffices to establish the result:
initially W =
1. if (T,lT ) ≤K (T ,lT ) because (T,lT ) ≤K (S ,lT ) with S an
immediate subtree of T , then W is updated by adding the
collection of nodes in the subgraph of T induced by
V (T )V (S );
→
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25. The proof, part II
→ Proof. (ii)
2. if (T,lT ) ≤K (T ,lT ) because lT (rT ) ≤A lT (rT ) and there is ξ
injective mapping the immediate subtrees of T to the immediate
subtrees of T such that (S,lT ) ≤K ξ(S,lT ), then [rT ] is the union
of W and the collection of nodes of the subgraph of T composed
by the immediate subtrees of T not in the image of ξ. Then,
inductively, the equivalence classes of the roots of the subtrees are
constructed, restarting with W = .
The equivalence classes [x] form a partition on V (T ), and thus a
c-equivalence ∼ as it is immediate to verify; moreover, there is an
evident injective function from V (T) to V (T )/∼ which maps the
root of each subtree in T into the root of some subtree in T . Finally,
labels are trivially preserved. Thus, since R(A) is a well quasi order by
Proposition 17, also T(A) is a well quasi order by Fact 11.
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26. An unsatisfactory theorem
So, Kruskal’s Theorem on pointed trees is extended to Kruskal’s
Theorem on trees as graphs. The key of the proof is that
(T,lT ) ≤K (T ,lT ) implies (T,lT ) ≤A
M
(T ,lT ), i.e., ≤K ⊆ ≤A
M.
Since ≤A
M extends ≤K, every bad sequence which happens to exist in
the collection of trees as graphs, is bad also in the collection of
pointed trees, for any choice of roots.
The same result holds for any quasi order extending ≤K. So, what
makes ≤A
M special? Why is the statement using ≤A
M referred to as a
Theorem? Does it depends only because it is useful?
The general answer in Mathematics is that something is useful
because it has a ‘good’ structure. And this is the case also for
Kruskal’s result.
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27. An alternative proof
Definition 26 (Node ordering)
Let (T,l) be a pointed tree, with r ∈ V (T) its root. If x,y ∈ V (T)
then x ≤T y when r y = (x y)◦(r x).
It is worth remarking that r x has to be a path, so it cannot
contain the same node twice, except for the endpoints. This fact
imposes a direction to the edges: x ≤T y when there is a path x y
which ‘goes only down’, thus y is ‘below’ x in the tree, or x is ‘closer’
than y to the root.
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28. An alternative proof
Definition 27 (Embedding via node ordering)
If (T,lT ) and (T ,lT ) are two pointed trees with labels over the quasi
order A = 〈A;≤A〉, then (T,lT ) ≤K (T ,lT ) when there is
ξ: V (T) → V (T ) injective such that:
if x ≤T y then ξ(x) ≤T ξ(y), ξ preserves the node ordering of T;
lT (x) ≤A lT (ξ(x)) for each x ∈ V (T).
Comparing with Definition 4, it immediately follows that
Fact 28
≤K=≤K.
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29. An alternative proof
Theorem 29 (Kruskal)
Let R (A) be the collection of pointed trees over A = 〈A;≤A〉. If A is
a well quasi order, then R (A) = R (A);≤K is a well quasi order.
Proof.
Following the proof of Proposition 17, consider a bad sequence
(Ti ,li ) i∈ω in R (A) minimising E (_) , define ∆ as before, thus
D = i∈ω ∆(Ti ,li );≤K is a well quasi order, and by Dickson’s
Lemma 14, D×D is a well quasi order. Thus, by the same argument
in Proposition 17, an injective ξ: V (Tn) → V (Tm) preserving the
node ordering of Tn and its labels can be found, for some n < m, thus
showing that Tn ≤KTm and contradicting (Ti ,li ) i∈ω to be bad.
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30. An alternative proof
Definition 30
Let R (A) the collection of pointed trees over A; define an
equivalence relation ≈ on R (A) such that (T,lT ) ≈ (T ,lT ) if and
only if V (T) = V (T ), E (T) = E (T ) and lT = lT , i.e., if the pointed
trees differ only by the choice of the root. Consider ≤≈
K, with
[(T,lT )] ≤≈
K
[(T ,lT )] if there are rT ∈ V (T), rT ∈ V (T ) such that
(T,lT ) ≤K (T ,lT ) as pointed trees with roots rT and rT respectively.
Fact 31
Each equivalence class [_]≈ denotes a non-pointed tree, that is,
R (A)/ ≈ is isomorphic to T (A).
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31. An alternative proof
This suggests that also the order relations on R(A)/ ≈ and T (A),
i.e., ≤≈
K and ≤A
M, may be related. The following result shows the
connection between the order relation on pointed trees and the graph
minor.
Proposition 32
If (T,lT ) and (T ,lT ) are trees, then [(T,lT )] ≤≈
K
[(T ,lT )] if and
only if (T,lT ) ≤A
M
(T ,lT ). Thus R(A)/≈ = R (A)/ ≈;≤≈
K is a quasi
order.
Kruskal’s Theorem follows because by Proposition 29 and
Proposition 13 R(A)/≈ is a well quasi order, and by Proposition 32,
R(A)/≈ ∼= T(A).
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32. An alternative proof
Proposition 32 really explains the Kruskal’s Theorem: the graph
minor relation ≤A
M is not some arbitrary extension of ≤K; rather, ≤A
M
is the relation obtained by forgetting the direction a choice of some
root imposes on a tree.
In other words, the collection of finite trees is a well quasi order with
respect to ≤A
M because each tree summarises a set of pointed trees,
differing only by the node which acts as a root, and, in turn, ≤A
M
summarises via the obvious quotient the relation ≤K which preserves
the structure of trees and their roots, the last emphasised bit being
what is abstracted away.
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33. The end
Questions?
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