Graph Minors, Bidimensionality and Algorithms

. FOMIN

FEDOR V

PART I Warsaw, 2010 Outline of the lectures

I Part I: Graph Minors

WQO

Kraskul’s theorem

Robertson-Seymour Graph Minors Theorem

Obstructions

Meta-algorithmic consequences Outline of the lectures

I Part II: Bidimensionality

Branch-width

Dynamic programming

Excluding Planar grid

Parameterized Algorithms Outline of the lectures

I Part III: Bidimensionality

Sphere-cut decompositions and Catalan structure

EPTAS

Kernelization (if time allows) PART I

Wagner’s Conjecture and Graph Minors series I Relation “ ” is quasi-ordering on X if it is reﬂexive and ≤ transitive

I X is well-quasi-ordered by if for every inﬁnite sequence ≤ x = x , x ,... in X, i < j, s.t. x x . 0 1 ∃ i ≤ j

I (xixj): a good pair of x . Sequence x is a good sequence.

W.Q.O. deﬁnition

I X be a collection of mathematical objects I X is well-quasi-ordered by if for every inﬁnite sequence ≤ x = x , x ,... in X, i < j, s.t. x x . 0 1 ∃ i ≤ j

I (xixj): a good pair of x . Sequence x is a good sequence.

W.Q.O. deﬁnition

I X be a collection of mathematical objects

I Relation “ ” is quasi-ordering on X if it is reﬂexive and ≤ transitive I (xixj): a good pair of x . Sequence x is a good sequence.

W.Q.O. deﬁnition

I X be a collection of mathematical objects

I Relation “ ” is quasi-ordering on X if it is reﬂexive and ≤ transitive

I X is well-quasi-ordered by if for every inﬁnite sequence ≤ x = x , x ,... in X, i < j, s.t. x x . 0 1 ∃ i ≤ j W.Q.O. deﬁnition

I X be a collection of mathematical objects

I Relation “ ” is quasi-ordering on X if it is reﬂexive and ≤ transitive

I X is well-quasi-ordered by if for every inﬁnite sequence ≤ x = x , x ,... in X, i < j, s.t. x x . 0 1 ∃ i ≤ j

I (xixj): a good pair of x . Sequence x is a good sequence. Proposition

A quasi-ordering X is a well-quasi-ordering if and only if X

contains

I neither inﬁnite antichain

I nor strictly decreasing sequence x0 > x1 > . . . .

PROOF: Ramsey arguments v 1. \v: vertex removal: e 2. \e: edge removal: v 3. /v: deg-2 vertex dissolution: e 4. /e: edge contraction

Let L ⊆ {\v, \e, /v, /e}

H ≤L G if H can be obtained from G after a sequence of operations in L

We deﬁne4 local operations: e 2. \e: edge removal: v 3. /v: deg-2 vertex dissolution: e 4. /e: edge contraction

Let L ⊆ {\v, \e, /v, /e}

H ≤L G if H can be obtained from G after a sequence of operations in L

We deﬁne4 local operations:

v 1. \v: vertex removal: v 3. /v: deg-2 vertex dissolution: e 4. /e: edge contraction

Let L ⊆ {\v, \e, /v, /e}

H ≤L G if H can be obtained from G after a sequence of operations in L

We deﬁne4 local operations:

v 1. \v: vertex removal: e 2. \e: edge removal: e 4. /e: edge contraction

Let L ⊆ {\v, \e, /v, /e}

H ≤L G if H can be obtained from G after a sequence of operations in L

We deﬁne4 local operations:

v 1. \v: vertex removal: e 2. \e: edge removal: v 3. /v: deg-2 vertex dissolution: Let L ⊆ {\v, \e, /v, /e}

H ≤L G if H can be obtained from G after a sequence of operations in L

We deﬁne4 local operations:

v 1. \v: vertex removal: e 2. \e: edge removal: v 3. /v: deg-2 vertex dissolution: e 4. /e: edge contraction H ≤L G if H can be obtained from G after a sequence of operations in L

We deﬁne4 local operations:

v 1. \v: vertex removal: e 2. \e: edge removal: v 3. /v: deg-2 vertex dissolution: e 4. /e: edge contraction

Let L ⊆ {\v, \e, /v, /e} We deﬁne4 local operations:

v 1. \v: vertex removal: e 2. \e: edge removal: v 3. /v: deg-2 vertex dissolution: e 4. /e: edge contraction

Let L ⊆ {\v, \e, /v, /e}

H ≤L G if H can be obtained from G after a sequence of operations in L Relation Notation \v \e /v /e WQO

induced subgraph (H ⊆in G) • NO

subgraph (H ⊆sb G) • • NO

spanning subgraph (H ⊆sp G) • NO induced topological minor (H ≤it G) • • NO

topological minor (H ≤tp G) • • • NO

induced minor (H ≤in G) • • NO

contraction (H ≤cn G) • NO

minor (H ≤mn G) • • • YES The fact that graphs are WQO by the minor relation was known as The Wagner’s Conjecture formulated by Klaus Wagner in the 1930s (?). The Graph Minors Series

The conjecture was proven by Neil Robertson and Paul Seymour in

their Graph Minor series of papers.

Now it is known as the Robertson & Seymour Theorem.

Width of the proof: < 10 cm (23 papers) Kruskal’s theorem

Theorem (Kruskal, 1960)

Trees are W.Q.O. by the topological minor relation.

(Formerly also known as the V´azsonyiconjecture) Proof of Kruskal’s theorem

We prove the following stronger statement:

I Rooted trees are W.Q.O. by the topological minor containment

Trees T and T 0 with roots r and r0. T T if isomorphism ϕ from some subdivision of T to a subtree ≤ 0 ∃ of T 0 preserving the tree-order on V (T ) associated with r and T . Proof of Kruskal’s theorem

We prove the following stronger statement:

I Rooted trees are W.Q.O. by the topological minor containment

r φ(r) Proof of Kruskal’s theorem

We prove the following stronger statement:

I Rooted trees are W.Q.O. by the topological minor containment

r φ(r) I Choose Tn to be the minimum order rooted tree such that there

is a bad sequence starting with T0,T1,T2,...,Tn

I (Tn)n 0 is a bad sequence ≥

Proof sketch:

Suppose that there is a bad sequence of rooted trees. For n 0 ≥ we select inductively the following bad sequence:

I Assume we constructed a sequence T0,T1,T2,...,Tn 1 s.t. − there is a bad sequence starting with T0,T1,T2,...,Tn 1 − I (Tn)n 0 is a bad sequence ≥

Proof sketch:

Suppose that there is a bad sequence of rooted trees. For n 0 ≥ we select inductively the following bad sequence:

I Assume we constructed a sequence T0,T1,T2,...,Tn 1 s.t. − there is a bad sequence starting with T0,T1,T2,...,Tn 1 − I Choose Tn to be the minimum order rooted tree such that there is a bad sequence starting with T0,T1,T2,...,Tn Proof sketch:

Suppose that there is a bad sequence of rooted trees. For n 0 ≥ we select inductively the following bad sequence:

I (Tn)n 0 is a bad sequence ≥ Let A1,A2,... be a sequence of rooted trees in A k k Let A be such that the tree Ti where A comes from has the smallest possible index.

k k+1 Then the sequence T0,...,Ti−1,A ,A ,..., is good (by (*)).

k0 k00 00 0 Then a comparable pair is of the form A tp A for k > k k. ≤ ≥

I For (Tn)n≥0 = T0,T1,T2,...

I An the set of components Tn rn. − S I = An A n≥0

r0 r1 r2 r3 r4

We ﬁrst prove that is WQO. I A Choose k with the smallest n(k).

k k+1 Then the sequence T0,...,Tn(k) 1,T ,T ,... is good (by − minimality of T and because T k T ). n(k) ⊂ n(k)

Proof that is WQO A

r0 r1 r2 r3 r4 Tk

k 1 2 Let (T )k 0 = T ,T ,... be a sequence of rooted trees in ≥ A For k deﬁne n(k) to be the minimum n such that T k A . ∈ n k k+1 Then the sequence T0,...,Tn(k) 1,T ,T ,... is good (by − minimality of T and because T k T ). n(k) ⊂ n(k)

Proof that is WQO A

r0 r1 r2 r3 r4 Tk

k 1 2 Let (T )k 0 = T ,T ,... be a sequence of rooted trees in ≥ A For k deﬁne n(k) to be the minimum n such that T k A . ∈ n Choose k with the smallest n(k). Proof that is WQO A

r0 r1 r2 r3 r4 Tk

k 1 2 Let (T )k 0 = T ,T ,... be a sequence of rooted trees in ≥ A For k deﬁne n(k) to be the minimum n such that T k A . ∈ n Choose k with the smallest n(k).

k k+1 Then the sequence T0,...,Tn(k) 1,T ,T ,... is good (by − minimality of T and because T k T ). n(k) ⊂ n(k) Thus it has a good pair (T,T 0)

Proof that is WQO A

r0 r1 r2 r3 r4 Tk

k k+1 The sequence T0,...,Tn(k) 1,T ,T ,... is good − Proof that is WQO A

r0 r1 r2 r3 r4 Tk

k k+1 The sequence T0,...,Tn(k) 1,T ,T ,... is good − Thus it has a good pair (T,T 0) Proof that is WQO: T T0,...,Tn(k) 1 ProofProof that thatis WQOis WQOA 6∈ − A A ProofProof that thatis WQOis WQO A A

r0 r1 r2 r3 r4 r0 r1 Tk r2 r3 r4 r0 r1 r2 r3 r4 r r r r r Tk 0 1 Tk 2 3 4 A Tk A A A k k+1 The sequence T0,...,Tn(k) 1, T ,T ,... is good k k+1 − The sequence T0,...,Tn(k) 1, T ,T ,... is good Good pair− cannot be Thus it has a good pair (T,T "k) k+1 Thus it has a good pair (T,T "k) k+1 The sequence T0,...,Tn(k) 1, T ,T ,... is good The sequence T0,...,Tn(k) 1, T ,T ,... is good − in a bad− sequence Thus it has a good pair (T,T ") Thus it has a good pair (T,T ") Thus it has a good pair (T,T ") ? Thus it has a good pair (T,T ") Thus it has a good pair (T,T ) Thus it has a good pair (T,T ") " ProofProof that thatis WQOis WQO A A ProofProof that thatis WQOis WQO Proof that is WQO: T T0,...,TA A n(k) 1 A 6∈ −

r0 r1 r2 r3 r4 r0 r1 Tk r2 r3 r4 r0 r1 r2 r3 r4 T Tk r0 r1 k r2 r3 r4 A Tk A A k k+1 A The sequence T0,...,Tn(k) 1, T ,T ,... is good − k k+1 The sequence T0,...,Tn(k) 1, T ,T ,... is good − Thus it has a good pair (T,T ")k k+1 By the The sequence T0,...,Tn(k) 1, T ,T ,... is good − Thus it has a good pair (T,T ")k k+1 Thus it has a good pair (T,T ") The sequence T0,...,Tn(k) 1, T ,T ,... is good choice− of Thus it has a good pair (T,T ") ? Thus it has a good pair (T,T ") Thus it has a good pair (T,T ")n(k) Thus it has a good pair (T,T ") Thus it has a good pair (T,T ")

If T = T = T = T i T , j < k i, then because j 0 ≤ n(i) ≤ n(i) n(k), we have T T . ≥ j ≤ n(i) Proof that is WQO: A

k k+1 T0,...,Tn(k) 1,T ,T ,... −

k k+1 I Thus both T 0 and T are from T ,T ,....

I Hence, is WQO. A Proof of Kruskal Theorem, contd.:

r0 r1 r2 r3 r4

I We know that is WQO. A <ω I next step, show that [ ] , the set of all ﬁnite subsets of , A A is WQO. Proof of Kruskal Theorem, contd.:

<ω I [ ] , the set of all ﬁnite subsets of , is WQO. A A

I For sets A, B , A B if there is an injective mapping ∈ A ≤ f : A B s.t. a f(a), a A → ≤ ∀ ∈

Lemma If , is WQO then so is [ ]<ω A A Proof of Kruskal Theorem, contd.:

Suppose (for a minute) that Lemma“ is WQO then so is [ ]<ω” A A holds. Then [ ]<ω is WQO. A <ω I The sequence (An)n 0 in [ ] should have a good pair ≥ A (A ,A ): A A , i < j i j i ≤ j x x x I injective mapping f : A A s.t. T f(T ), T A ∃ i → j ≤ ∀ ∈ i

ri rj

Ti Tj

A1 A2 A3 A4 A5 A10 A20 A30 A40 A50 A60 Proof of Kruskal Theorem, contd.:

Extend f to ϕ by ϕ(ri) = ϕ(rj)

I T T i ≤ j

I (Ti,Tj) is a good pair in a bad sequence - CONTRADICTION!

ri rj

Ti Tj

A1 A2 A3 A4 A5 A10 A20 A30 A40 A50 A60 For every i 1 choose A s.t. A ,A ,...,A is a bad sequence ≥ i 0 1 i and A min | i| →

What remains

Lemma If is WQO then so is [ ]<ω A A PROOF: Assume that is WQO but [ ]<ω not A A What remains

Lemma If is WQO then so is [ ]<ω A A PROOF: Assume that is WQO but [ ]<ω not A A For every i 1 choose A s.t. A ,A ,...,A is a bad sequence ≥ i 0 1 i and A min | i| → pick a A , and set B = A a . n ∈ n n n \ n (an)n 0 is a good sequence, pick up an inﬁnite increasing ≥

subsequence (ani )i 0 ≥

What remains

PROOF: Assume that is WQO but [ ]<ω not A A For every i 1 choose A s.t. A ,A ,...,A is a bad sequence ≥ i 0 1 i and A min | i| → (an)n 0 is a good sequence, pick up an inﬁnite increasing ≥

subsequence (ani )i 0 ≥

What remains

PROOF: Assume that is WQO but [ ]<ω not A A For every i 1 choose A s.t. A ,A ,...,A is a bad sequence ≥ i 0 1 i and A min | i| → pick a A , and set B = A a . n ∈ n n n \ n What remains

PROOF: Assume that is WQO but [ ]<ω not A A For every i 1 choose A s.t. A ,A ,...,A is a bad sequence ≥ i 0 1 i and A min | i| → pick a A , and set B = A a . n ∈ n n n \ n (an)n 0 is a good sequence, pick up an inﬁnite increasing ≥

subsequence (ani )i 0 ≥ This is a good sequence because of minimality of A0,A1,...

Good pair cannot be of the form (Ai,Aj) and of the form (A ,B A ) i j ≤ j But good pair (B ,B ) and a a imply that i j i ≤ j (A = B , a ,A = B a ) is a good pair. i i ∪ i j j ∪ j

What remains

PROOF: take

A0,...,An0 1,Bn0 ,Bn1 ,Bn2 ,... − Good pair cannot be of the form (Ai,Aj) and of the form (A ,B A ) i j ≤ j But good pair (B ,B ) and a a imply that i j i ≤ j (A = B , a ,A = B a ) is a good pair. i i ∪ i j j ∪ j

What remains

PROOF: take

A0,...,An0 1,Bn0 ,Bn1 ,Bn2 ,... −

This is a good sequence because of minimality of A0,A1,... But good pair (B ,B ) and a a imply that i j i ≤ j (A = B , a ,A = B a ) is a good pair. i i ∪ i j j ∪ j

What remains

PROOF: take

A0,...,An0 1,Bn0 ,Bn1 ,Bn2 ,... −

This is a good sequence because of minimality of A0,A1,...

Good pair cannot be of the form (Ai,Aj) and of the form (A ,B A ) i j ≤ j What remains

PROOF: take

A0,...,An0 1,Bn0 ,Bn1 ,Bn2 ,... −

This is a good sequence because of minimality of A0,A1,...

Good pair cannot be of the form (Ai,Aj) and of the form (A ,B A ) i j ≤ j But good pair (B ,B ) and a a imply that i j i ≤ j (A = B , a ,A = B a ) is a good pair. i i ∪ i j j ∪ j Friedman (2002) observed that Kruskal’s theorem has special cases

that can be stated but not proved in ﬁrst-order arithmetic.

(Though they can easily be proved in second-order arithmetic.)

REMARKS

The proof is non-constructive. REMARKS

The proof is non-constructive.

Friedman (2002) observed that Kruskal’s theorem has special cases

that can be stated but not proved in ﬁrst-order arithmetic.

(Though they can easily be proved in second-order arithmetic.) How can we go further than trees?

We have to deﬁne the tree-likeness of a graph. Treewidth of a graph

Theorem (Robertson & Seymour, GM IV)

For every k 0, graphs of treewidth at most k are WQO. ≥ We will be back to this theorem soon...

What if the treewidth is unbounded?

Start from planar graphs.

Theorem (Robertson & Seymour, GM IV)

There is function f such that for every k 0, a planar graph G of ≥ treewidth at least f(k) contains k as a minor. What if the treewidth is unbounded?

Start from planar graphs.

Theorem (Robertson & Seymour, GM IV)

There is function f such that for every k 0, a planar graph G of ≥ treewidth at least f(k) contains k as a minor.

We will be back to this theorem soon... What if the treewidth is unbounded?

Observation: Every planar graph G on n vertices is a minor of suﬃciently large grid. If G , i 1, is of treewidth more than f(G ), it has large grid I i ≥ 0 as a minor, which contains G as a minor. Thus G G . 0 0 ≤ i I All graphs in G1,G2 ... should have

treewidth at most c = f(V (G0)), and thus are WQO!

Wagner’s conjecture for planar graphs, GM IV

I Let G0,G1,..., be a bad sequence of planar graphs. I All graphs in G1,G2 ... should have

treewidth at most c = f(V (G0)), and thus are WQO!

Wagner’s conjecture for planar graphs, GM IV

I Let G0,G1,..., be a bad sequence of planar graphs.

If G , i 1, is of treewidth more than f(G ), it has large grid I i ≥ 0 as a minor, which contains G as a minor. Thus G G . 0 0 ≤ i Wagner’s conjecture for planar graphs, GM IV

I Let G0,G1,..., be a bad sequence of planar graphs.

If G , i 1, is of treewidth more than f(G ), it has large grid I i ≥ 0 as a minor, which contains G as a minor. Thus G G . 0 0 ≤ i I All graphs in G1,G2 ... should have

treewidth at most c = f(V (G0)), and thus are WQO! Is it again “all about trees”?

Answer: YES andNO!

NO: We should now also deal with graphs embedded in surfaces!

YES: But surfaces that are arranged together as trees!

What about non-planar graphs?

How graphs excluding a non-planar graph look like? Answer: YES andNO!

NO: We should now also deal with graphs embedded in surfaces!

YES: But surfaces that are arranged together as trees!

What about non-planar graphs?

How graphs excluding a non-planar graph look like?

Is it again “all about trees”? NO: We should now also deal with graphs embedded in surfaces!

YES: But surfaces that are arranged together as trees!

What about non-planar graphs?

How graphs excluding a non-planar graph look like?

Is it again “all about trees”?

Answer: YES andNO! YES: But surfaces that are arranged together as trees!

What about non-planar graphs?

How graphs excluding a non-planar graph look like?

Is it again “all about trees”?

Answer: YES andNO!

NO: We should now also deal with graphs embedded in surfaces! What about non-planar graphs?

How graphs excluding a non-planar graph look like?

Is it again “all about trees”?

Answer: YES andNO!

NO: We should now also deal with graphs embedded in surfaces!

YES: But surfaces that are arranged together as trees! Further reading

I Chapter 12 of Graph Theory, R. Diestel, 3rd edition. Algorithmic consequences of the R&S theorem A graph parameter is minor-closed if H mn G p(H) p(G) ≤ ⇒ ≤

A graph parameter p is a function mapping graphs to positive integers. A graph parameter p is a function mapping graphs to positive integers.

A graph parameter is minor-closed if H mn G p(H) p(G) ≤ ⇒ ≤ Each parameter p corresponds to a parameterized problem:

p-Problem of Deciding p

Instance: a graph G and an integer k 0. ≥ Parameter: k

Question: p(G) k? ≤ I Not all parameterized problems admit FPT-algorithms. There are parameterized complexity classes like W[1], W[2], or W[P] and

adequate reductions such that when a problem is hard for them is not

expected to have an FPT-algorithm.

We say that a parameterized (by k) problem is FPT (ﬁxed parameter tractable) if it can be solved in time

O(f(k) nO(1)) steps ·

(n is the size of the input, f depends only one the parameter k.) There are parameterized complexity classes like W[1], W[2], or W[P] and

adequate reductions such that when a problem is hard for them is not

expected to have an FPT-algorithm.

We say that a parameterized (by k) problem is FPT (ﬁxed parameter tractable) if it can be solved in time

O(f(k) nO(1)) steps ·

(n is the size of the input, f depends only one the parameter k.)

I Not all parameterized problems admit FPT-algorithms. We say that a parameterized (by k) problem is FPT (ﬁxed parameter tractable) if it can be solved in time

O(f(k) nO(1)) steps ·

(n is the size of the input, f depends only one the parameter k.)

I Not all parameterized problems admit FPT-algorithms. There are parameterized complexity classes like W[1], W[2], or W[P] and adequate reductions such that when a problem is hard for them is not expected to have an FPT-algorithm. Minor-closed parameters:

I vertex cover, vc(G) I feedback vertex set, fvs(G) I branchwidth, bw(G) I minimum maximal matching, mmm(G)

k-almostΠ(G) = min S G S Π (Π is any minor-closed class) I {| | | − ∈ } I the genus of a graph, γ(G)

p(G) = min k Pk mn G I { | 6≤ } Consequence of R&S Theorem: for any minor-closed graph class the I G set of graphs not in it have a ﬁnite set of minor-minimal elements. we denote this set ob( ) and we call it obstruction set of . I G G Observe: if is minor-closed then ob( ) is ﬁnite. I G G Examples of obstruction sets

I Trees: K3

I Outerplanar Graphs: K2,3, K4

I bw(G) ≤ 2: K4

I planar graphs: K3,3, K5 (Theorem Kuratowski-Понтрягин)

I link-free graphs:7 graphs (Petersen family: X-Y transformations of K6) Examples of obstruction sets

I Graphs with a vertex cover of size ≤ 5: 56 graphs I Graphs with a vertex cover of size ≤ 6: 260 graphs I Graphs with a feedback vertex of size ≤ 3: ≥ 744 graphs I Graphs embeddable in the projective plane: 35 graphs

I Graphs embeddable in the torus: ≥ 2200 graphs k I Graphs with branchwidth ≤ k graphs: obstructions have size ≤ (6 − 1)/5. [Geelen, Gerards, Robertson, and Whittlee, JCTSB’ 03]

I Graphs with a vertex cover ≤ k graphs: obstructions have size ≤ 2(k + 1). [Michael J. Dinneen, Rongwei Lai, Disc. Math, ’07]

Upper Bounds? I Graphs with a vertex cover ≤ k graphs: obstructions have size ≤ 2(k + 1). [Michael J. Dinneen, Rongwei Lai, Disc. Math, ’07]

Upper Bounds?

k I Graphs with branchwidth ≤ k graphs: obstructions have size ≤ (6 − 1)/5. [Geelen, Gerards, Robertson, and Whittlee, JCTSB’ 03] Upper Bounds?

k I Graphs with branchwidth ≤ k graphs: obstructions have size ≤ (6 − 1)/5. [Geelen, Gerards, Robertson, and Whittlee, JCTSB’ 03]

I Graphs with a vertex cover ≤ k graphs: obstructions have size ≤ 2(k + 1). [Michael J. Dinneen, Rongwei Lai, Disc. Math, ’07] 2 I Searching an active and visible (or pathwidth) ≤ k: ≥ (k!) graphs. [Parsons ’98], [Ellis, Sudborough, &Turner ’94], [Takahashi, Ueno, & Kajitani, 94]

Ω(k·log k) I Searching an active and visible (or treewidth) ≤ k: 2 graphs.

[Arvind Gupta, Damon Kaller, and Thomas Shermer, ICALP’99]

Lower Bounds? Ω(k·log k) I Searching an active and visible (or treewidth) ≤ k: 2 graphs.

[Arvind Gupta, Damon Kaller, and Thomas Shermer, ICALP’99]

Lower Bounds?

2 I Searching an active and visible (or pathwidth) ≤ k: ≥ (k!) graphs. [Parsons ’98], [Ellis, Sudborough, &Turner ’94], [Takahashi, Ueno, & Kajitani, 94] Lower Bounds?

2 I Searching an active and visible (or pathwidth) ≤ k: ≥ (k!) graphs. [Parsons ’98], [Ellis, Sudborough, &Turner ’94], [Takahashi, Ueno, & Kajitani, 94]

Ω(k·log k) I Searching an active and visible (or treewidth) ≤ k: 2 graphs.

[Arvind Gupta, Damon Kaller, and Thomas Shermer, ICALP’99] Main meta-algorithmic consequence of GM

Theorem If p is a minor-closed parameter, then p-Problem of Deciding

p is in FPT by an O(f(k) n3) step algorithm. ·

Moreover, for planar inputs (and more) the above algorithms are linear. k, k is minor-closed and its obstruction set ob( k) is ﬁnite I ∀ G G Let ob( k) has at most f(k) elements of size at most g(k). I G G k iﬀ H mn G I ∈ G ∀H∈ob(Gk) 6≤

Proof:

Let k be the graph class of the YES instances of I G p-Problem of Deciding p with k as parameter. Let ob( k) has at most f(k) elements of size at most g(k). I G G k iﬀ H mn G I ∈ G ∀H∈ob(Gk) 6≤

Proof:

Let k be the graph class of the YES instances of I G p-Problem of Deciding p with k as parameter.

k, k is minor-closed and its obstruction set ob( k) is ﬁnite I ∀ G G G k iﬀ H mn G I ∈ G ∀H∈ob(Gk) 6≤

Proof:

Let k be the graph class of the YES instances of I G p-Problem of Deciding p with k as parameter.

k, k is minor-closed and its obstruction set ob( k) is ﬁnite I ∀ G G Let ob( k) has at most f(k) elements of size at most g(k). I G Proof:

Let k be the graph class of the YES instances of I G p-Problem of Deciding p with k as parameter.

k, k is minor-closed and its obstruction set ob( k) is ﬁnite I ∀ G G Let ob( k) has at most f(k) elements of size at most g(k). I G G k iﬀ H mn G I ∈ G ∀H∈ob(Gk) 6≤ An algorithm for p-Problem of Deciding p.

Decide-p(G,k )

1. for all H ∈ ob(Gk)

3 2. check (in O(h(k) · n ) steps) whether H ≤mn G 3. if the answer if YES, then output NO 4. output YES. Notice that, for each k, the class of YES-instances

for p-Feedback Vertex Set is minor-closed:

yes = G S V (G)[ k],K G S I Gk { | ∃ ⊆ ≤ 3 6≤mn \ } Therefore: p-Feedback Vertex Set FPT I ∈

Applications

p-Feedback Vertex Set

Instance: a graph G and a positive integer k.

Parameter: k

Question: does G contain a set k meeting all cycles of G? yes = G S V (G)[ k],K G S I Gk { | ∃ ⊆ ≤ 3 6≤mn \ } Therefore: p-Feedback Vertex Set FPT I ∈

Applications

p-Feedback Vertex Set

Instance: a graph G and a positive integer k.

Parameter: k

Question: does G contain a set k meeting all cycles of G?

Notice that, for each k, the class of YES-instances

for p-Feedback Vertex Set is minor-closed: Therefore: p-Feedback Vertex Set FPT I ∈

Applications

p-Feedback Vertex Set

Instance: a graph G and a positive integer k.

Parameter: k

Question: does G contain a set k meeting all cycles of G?

Notice that, for each k, the class of YES-instances

for p-Feedback Vertex Set is minor-closed:

yes = G S V (G)[ k],K G S I Gk { | ∃ ⊆ ≤ 3 6≤mn \ } Applications

p-Feedback Vertex Set

Instance: a graph G and a positive integer k.

Parameter: k

Question: does G contain a set k meeting all cycles of G?

Notice that, for each k, the class of YES-instances

for p-Feedback Vertex Set is minor-closed:

yes = G S V (G)[ k],K G S I Gk { | ∃ ⊆ ≤ 3 6≤mn \ } Therefore: p-Feedback Vertex Set FPT I ∈ Notice that, for each k, the class yes of YES-instances Gk for p-Genus is minor-closed:

Therefore: p-Genus FPT I ∈

p-Genus

Instance: a graph G and a positive integer k.

Parameter: k

Question: is there an embedding of G in a surface of

(orientable) genus k? ≤ for p-Genus is minor-closed:

Therefore: p-Genus FPT I ∈

p-Genus

Instance: a graph G and a positive integer k.

Parameter: k

Question: is there an embedding of G in a surface of

(orientable) genus k? ≤ Notice that, for each k, the class yes of YES-instances Gk Therefore: p-Genus FPT I ∈

p-Genus

Instance: a graph G and a positive integer k.

Parameter: k

Question: is there an embedding of G in a surface of

(orientable) genus k? ≤ Notice that, for each k, the class yes of YES-instances Gk for p-Genus is minor-closed: p-Genus

Instance: a graph G and a positive integer k.

Parameter: k

Question: is there an embedding of G in a surface of

(orientable) genus k? ≤ Notice that, for each k, the class yes of YES-instances Gk for p-Genus is minor-closed:

Therefore: p-Genus FPT I ∈ Notice that, for each k, the class of YES-instances of p-Mutual

Linking is minor-closed:

Therefore: p-Mutual Linking FPT I ∈

p-Mutual Linking

Instance: a graph G and a positive integer k.

Parameter: k

Question: is there an embedding of G in R3 such that no more than pairs k cycles are linked? Therefore: p-Mutual Linking FPT I ∈

p-Mutual Linking

Instance: a graph G and a positive integer k.

Parameter: k

Question: is there an embedding of G in R3 such that no more than pairs k cycles are linked?

Notice that, for each k, the class of YES-instances of p-Mutual

Linking is minor-closed: p-Mutual Linking

Instance: a graph G and a positive integer k.

Parameter: k

Question: is there an embedding of G in R3 such that no more than pairs k cycles are linked?

Notice that, for each k, the class of YES-instances of p-Mutual

Linking is minor-closed:

Therefore: p-Mutual Linking FPT I ∈ Notice that, for each k, the class of NO-instances

for p-Longest Cycle is minor-closed:

no = G C G I Gk { | k 6≤mn } Therefore: p-Longest Cycle FPT. I ∈

p-Longest Cycle

Instance: a graph G and a positive integer k.

Parameter: k

Question: does G contain a cycle of length k? for p-Longest Cycle is minor-closed:

no = G C G I Gk { | k 6≤mn } Therefore: p-Longest Cycle FPT. I ∈

p-Longest Cycle

Instance: a graph G and a positive integer k.

Parameter: k

Question: does G contain a cycle of length k?

Notice that, for each k, the class of NO-instances no = G C G I Gk { | k 6≤mn } Therefore: p-Longest Cycle FPT. I ∈

p-Longest Cycle

Instance: a graph G and a positive integer k.

Parameter: k

Question: does G contain a cycle of length k?

Notice that, for each k, the class of NO-instances for p-Longest Cycle is minor-closed: Therefore: p-Longest Cycle FPT. I ∈

p-Longest Cycle

Instance: a graph G and a positive integer k.

Parameter: k

Question: does G contain a cycle of length k?

Notice that, for each k, the class of NO-instances for p-Longest Cycle is minor-closed:

no = G C G I Gk { | k 6≤mn } p-Longest Cycle

Instance: a graph G and a positive integer k.

Parameter: k

Question: does G contain a cycle of length k?

Notice that, for each k, the class of NO-instances for p-Longest Cycle is minor-closed:

no = G C G I Gk { | k 6≤mn } Therefore: p-Longest Cycle FPT. I ∈ 2. we know obs( for few classes and for small values of k Gk) 3. when we have estimations of f(k) and g(k), they are

immense. The corresponding FPT-algorithms have a

heavy parameter dependance.

Drawbacks of graph minors meta-algorithm:

1. the above proof is non-constructive as we do not know

obs( Gk) 2. we know obs( for few classes and for small values of k Gk) 3. when we have estimations of f(k) and g(k), they are

immense. The corresponding FPT-algorithms have a

heavy parameter dependance.

Drawbacks of graph minors meta-algorithm:

1. the above proof is non-constructive as we do not know

obs( Gk) 3. when we have estimations of f(k) and g(k), they are

immense. The corresponding FPT-algorithms have a

heavy parameter dependance.

Drawbacks of graph minors meta-algorithm:

1. the above proof is non-constructive as we do not know

obs( Gk) 2. we know obs( for few classes and for small values of k Gk) Drawbacks of graph minors meta-algorithm:

1. the above proof is non-constructive as we do not know

obs( Gk) 2. we know obs( for few classes and for small values of k Gk) 3. when we have estimations of f(k) and g(k), they are

immense. The corresponding FPT-algorithms have a

heavy parameter dependance. Conclusion:

However spectacular such unexpected solutions to long-standing problems may be, viewing the graph minor theory merely in terms of its corollaries will not do its justice. At least as important are the techniques developed for its proof... Reinhard Diestel, Graph Theory, 4th edition. Otherwise, exploit the structure of the obstruction to the small

treewidth!

Conclusion:

Main motif of GM: If graph has a tree like structure (small treewidth), great! Conclusion:

Main motif of GM: If graph has a tree like structure (small treewidth), great!

Otherwise, exploit the structure of the obstruction to the small

treewidth! But it appeared that the WIN/WIN approach of GM: either small

treewidth or big obstruction is worth to try!

Conclusion:

From Algorithmic perspective:

Seems that the consequences of GM are great but completely

unpractical. Conclusion:

From Algorithmic perspective:

Seems that the consequences of GM are great but completely

unpractical.

But it appeared that the WIN/WIN approach of GM: either small

treewidth or big obstruction is worth to try! Further reading. Kruskal’s Theorem and Graph Minors

R. Diestel, Graph Theory, Third Edition, Springer, Chapter

12 Further reading. Algorithmic Applications of Graph Minors

R. Downey and M. Fellows, Parameterized Complexity,

Springer