Explaining the Kruskal Tree Theore

Explaining the Kruskal’s Tree Theorem

Dr M Benini, Dr R Bonacina

Università degli Studi dell’Insubria

Logic SeminarsJAIST,

May 12th, 2017

The theorem

Theorem 1 (Kruskal)The collection T (A) of all the finite trees labelled over a well quasiorder A ordered by homeomorphic embedding, is a well quasi orderT(A)= ⟨T (A);≤E ⟩.The T1 ≤E T2 relation means that there is an embedding η of T1 intoT2, i.e., a function which maps the nodes and the arcs of T1 intothose of T2 and preserves the structure of the T1 tree.It is worth noticing that the statement is not precise, since thedefinitions of ‘tree’ and ‘preserving the structure’ are left implicit.

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The theorem

In fact, there is an ambiguous point in the statement: it is usuallyintended as speaking of trees as some special graphs, with the notionof embedding captured via the graph minor relation, while it is provedby using an inductive definition of trees, close to the usual one inComputer Science, with a natural but ad-hoc notion of embedding.So, we are speaking of two distinct theorems, and about theirrelation. Also, we have many proofs of one of them, while the other isusually relegated to a footnote in the end, or a quick hint, or anexercise, when mentioned. But it is the unproved one which is reallyused in Mathematics.

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The theorem

Definition 2 (Tree)Let A= ⟨A;≤A⟩ be a quasi order. A tree T is inductively defined as1. 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 immediatesubtrees of the root.

A labelled tree (T , l) over A is a tree T and a function l from itsnodes 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, werefer to the trees as for the former one as pointed trees.( 4 of 33 )

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 ′) bepointed trees. Then (T , lT )≤K (T ′, lT ′) if and only if one of thefollowing conditions applies1. 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 theimmediate subtrees (S , lT ) of (T , lT ) to those (S ′, lT ′) of (T ′, lT ′)such that (S , lT )≤K (S ′, lT ′).

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Well quasi orders

Definition 5 (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 .

Intuitively, a quasi order is an order in which we admit two elementsto be equivalent but not equal.

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Well quasi orders

Definition 6 (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.

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Well quasi orders

Definition 7 (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 8 (Well quasi order)A well quasi order is a well founded quasi order having the finiteantichain property.

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Nash-Williams’s toolbox

Definition 9 (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.

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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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Fact 11Let A= ⟨A;≤⟩ be a well quasi order. Then, for every quasi orderA+ = ⟨

A;≤+⟩such that ≤⊆≤+, A+ is a well quasi order.

Fact 12Let A= ⟨A;≤A⟩ be a well quasi order. Then, every quasi orderB= ⟨B;≤B⟩ with B ⊆A and ≤B the restriction of ≤A to B, is a wellquasi order.

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Proposition 13Let ⟨A;≤⟩ be a well quasi order and let ≈ be an equivalence relationon A such that ≤≈ is a quasi ordering of A/≈, with [x ]≈ ≤≈ [y ]≈ if andonly if there are x ′ ∈ [x ]≈ and y ′ ∈ [y ]≈ such that x ′ ≤ y ′. Then⟨A/≈;≤≈

⟩is a well quasi order.

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Lemma 14 (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.

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Lemma 15Let 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).

So, if A is a quasi order, but not a well quasi order, then it contains abad sequence which is minimal with respect to some measure f andsome comparison criterion r , normally =.

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Let B= ⟨B;≤B⟩ be a quasi order. Let ⟨W ;≤⟩ be a total well foundedquasi 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 andminimal with respect to µ and = by Lemma 15.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 16Let D= ⟨⋃i>p∆(Bi);≤B⟩. Then D is a well quasi order.

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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 well knownresults, e.g., Dickson’s Lemma, it is usually easy to deduce that Clies 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.

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The proof

Theorem 17 (Kruskal)Let R (A) be the collection of pointed trees over A= ⟨A;≤A⟩. If A isa 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 isa 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, sop =max I 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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The proof

,→ Proof. (ii)For i > p, define ∆(Ti , li) as the set composed by the two connectedcomponents

{(T 1

i , li)

,(T 2

i , li)}

obtained deleting some arc{ri ,xi } ∈E (Ti): each component is a pointed tree having one endpointof {ri ,xi } as its root. We stipulate that the root of T 1

i is ri and theroot of T 2

i is xi . Clearly, if(T j

i , li)∈∆(Ti , li),

(T j

i , li)≤K (Ti , li) and∣∣∣E (

T ji , 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 wellquasi order. Considering the sequence

{((T 1

i , li)

,(T 2

i , li))}

i>p, byTheorem 10 there are m> n such that

(T 1

n , ln)≤K

(T 1

m, lm), thus

ln (rn)≤A lm (rm), and(T 2

n , ln)≤K

(T 2

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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Pointed trees versus graphs

Consider the following pair of incomparable trees, and decomposethem as in the previous proof:

6∥ but =

The decomposition yields two pairs of subtrees which are identical asgraphs but different as pointed trees.Thus, extending the proof of Kruskal’s Theorem to trees seen asgraphs is not immediate.

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Trees as graphs

Definition 18 (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

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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 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.

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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) forevery 0< i < j ≤ n, vi 6= vj .

Definition 22 (Connected graph)A graph is connected when there is at least one path between everypair of nodes.

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Trees as graphs

Definition 23 (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 24Let G be the collection of all the finite graphs.Then ⟨G ;≤S⟩ and ⟨G ;≤M⟩ are partial orders.

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The proof, part II

Theorem 25 (Kruskal)Let T (A) be the collection of all the pointed trees labelled overA= ⟨A;≤A⟩. If A is a well quasi order, then T(A)= ⟨

T (A);≤AM

⟩is a

well quasi order.

Proof. (i)Notice how (T , lT )≤K (T ′, lT ′) implies (T , lT )≤A

M (T ′, lT ′). In fact, asimple 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 thecollection of nodes in the subgraph of T ′ induced byV (T ′) \V (S ′);

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The proof, part II

,→ Proof. (ii)

2. if (T , lT )≤K (T ′, lT ′) because lT (rT )≤A lT ′ (rT ′) and there is ξinjective mapping the immediate subtrees of T to the immediatesubtrees of T ′ such that (S , lT )≤K ξ(S , lT ), then [rT ′ ] is the unionof W and the collection of nodes of the subgraph of T ′ composedby the immediate subtrees of T ′ not in the image of ξ. Then,inductively, the equivalence classes of the roots of the subtrees areconstructed, restarting with W =;.

The equivalence classes [x ] form a partition on V (T ′), and thus ac-equivalence ∼ as it is immediate to verify; moreover, there is anevident injective function from V (T ) to V (T ′)/∼ which maps theroot 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 byProposition 17, also T(A) is a well quasi order by Fact 11.

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An unsatisfactory theorem

So, Kruskal’s Theorem on pointed trees is extended to Kruskal’sTheorem 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 ⊆≤AM.

Since ≤AM extends ≤K, every bad sequence which happens to exist in

the collection of trees as graphs, is bad also in the collection ofpointed trees, for any choice of roots.The same result holds for any quasi order extending ≤K. So, whatmakes ≤A

M special? Why is the statement using ≤AM referred to as a

Theorem? Does it depends only because it is useful?The general answer in Mathematics is that something is usefulbecause it has a ‘good’ structure. And this is the case also forKruskal’s result.

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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 cannotcontain the same node twice, except for the endpoints. This factimposes a direction to the edges: x ≤T y when there is a path x ywhich ‘goes only down’, thus y is ‘below’ x in the tree, or x is ‘closer’than y to the root.

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Definition 27 (Embedding via node ordering)If (T , lT ) and (T ′, lT ′) are two pointed trees with labels over the quasiorder A= ⟨A;≤A⟩, then (T , lT )≤′

K (T ′, lT ′) when there isξ : 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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Theorem 29 (Kruskal)Let R (A) be the collection of pointed trees over A= ⟨A;≤A⟩. If A isa 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 argumentin Proposition 17, an injective ξ : V (Tn)→V (Tm) preserving thenode ordering of Tn and its labels can be found, for some n<m, thusshowing that Tn ≤′

KTm and contradicting{(Ti , li)

}i∈ω to be bad.

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Definition 30Let R (A) the collection of pointed trees over A; define anequivalence relation ≈ on R (A) such that (T , lT )≈ (T ′, lT ′) if andonly if V (T )=V (T ′), E (T )=E (T ′) and lT = lT ′ , i.e., if the pointedtrees 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 31Each equivalence class [_]≈ denotes a non-pointed tree, that is,R (A)/≈ is isomorphic to T (A).

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This suggests that also the order relations on R(A)/≈ and T (A),i.e., ≤≈

K and ≤AM, may be related. The following result shows the

connection between the order relation on pointed trees and the graphminor.Proposition 32If (T , lT ) and (T ′, lT ′) are trees, then [(T , lT )]≤≈

K [(T ′, lT ′)] if andonly 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 andProposition 13 R(A)/≈ is a well quasi order, and by Proposition 32,R(A)/≈∼=T(A).

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Proposition 32 really explains the Kruskal’s Theorem: the graphminor relation ≤A

M is not some arbitrary extension of ≤K; rather, ≤AM

is the relation obtained by forgetting the direction a choice of someroot imposes on a tree.In other words, the collection of finite trees is a well quasi order withrespect 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

Msummarises via the obvious quotient the relation ≤′

K which preservesthe structure of trees and their roots, the last emphasised bit beingwhat is abstracted away.

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The end

Questions?

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