Graph Decompositions and Well-Quasi-Ordering

Graph decompositions and well-quasi-ordering

Jean-Florent Raymond

LIRMM, University of Montpellier, France, and MIMUW, University of Warsaw, Poland

Orl´eans,November 2015

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 1 / 14 {0}, {1}, {2},... is an inﬁnite antichain wrt. ⊆ → (P(N), ⊆) is not a WQO; ? (A , 6subseq) with A ﬁnite: WQO; (graphs, 6minor): WQO.

1 1 1, 2 , 3 ,... is an inﬁnite decreasing sequence wrt. 6 → (Q, 6) is not a WQO;

Well-quasi-ordering

A WQO is an order where: every decreasing sequence is ﬁnite; every sequence of non-comparable elements is ﬁnite.

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 2 / 14 ? (A , 6subseq) with A ﬁnite: WQO; (graphs, 6minor): WQO.

{0}, {1}, {2},... is an inﬁnite antichain wrt. ⊆ → (P(N), ⊆) is not a WQO;

Well-quasi-ordering

A WQO is an order where: every decreasing sequence is ﬁnite; every sequence of non-comparable elements is ﬁnite.

Examples: 1 1 1, 2 , 3 ,... is an inﬁnite decreasing sequence wrt. 6 → (Q, 6) is not a WQO;

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 2 / 14 ? (A , 6subseq) with A ﬁnite: WQO; (graphs, 6minor): WQO.

Well-quasi-ordering

A WQO is an order where: every decreasing sequence is ﬁnite; every sequence of non-comparable elements is ﬁnite.

Examples: 1 1 1, 2 , 3 ,... is an inﬁnite decreasing sequence wrt. 6 → (Q, 6) is not a WQO; {0}, {1}, {2},... is an inﬁnite antichain wrt. ⊆ → (P(N), ⊆) is not a WQO;

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 2 / 14 Well-quasi-ordering

A WQO is an order where: every decreasing sequence is ﬁnite; every sequence of non-comparable elements is ﬁnite.

Examples: 1 1 1, 2 , 3 ,... is an infinite decreasing sequence wrt. 6 → (Q, 6) is not a WQO; {0}, {1}, {2},... is an infinite antichain wrt. ⊆ → (P(N), ⊆) is not a WQO; ? (A , 6subseq) with A finite: WQO; (graphs, 6minor): WQO.

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 2 / 14 Why do we like well-quasi-orders?

Upwards closed classes have a ﬁnite number of minimal elements.

m1 m3

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 3 / 14 Why do we like well-quasi-orders?

Upwards closed classes have a ﬁnite number of minimal elements.

b a

m1 m3

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 3 / 14 Why do we like well-quasi-orders?

Upwards closed classes have a ﬁnite number of minimal elements.

x ∈ U ⇐⇒ m1 6 x ∨ · · · ∨ mk 6 x (ﬁnite base)

m1 m3

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 3 / 14 Why do we like well-quasi-orders?

Upwards closed classes have a ﬁnite number of minimal elements.

x ∈ U ⇐⇒ m1 6 x ∨ · · · ∨ mk 6 x (ﬁnite base) Membership testing can be done in a ﬁnite number of checks.

m1 m3

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 3 / 14 Why do we like well-quasi-orders?

Upwards closed classes have a ﬁnite number of minimal elements.

x ∈ U ⇐⇒ m1 6 x ∨ · · · ∨ mk 6 x (ﬁnite base) Membership testing can be done in a ﬁnite number of checks.

m1 m3

m2 Downwards closed classes have a ﬁnite number of obstructions.

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 3 / 14 Why do we like well-quasi-orders?

Upwards closed classes have a ﬁnite number of minimal elements.

x ∈ U ⇐⇒ m1 6 x ∨ · · · ∨ mk 6 x (ﬁnite base) Membership testing can be done in a ﬁnite number of checks.

m1 m3

m2 Downwards closed classes have a ﬁnite number of obstructions. a

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 3 / 14 Why do we like well-quasi-orders?

Upwards closed classes have a ﬁnite number of minimal elements.

x ∈ U ⇐⇒ m1 6 x ∨ · · · ∨ mk 6 x (ﬁnite base) Membership testing can be done in a ﬁnite number of checks.

m1 m3

m2 Downwards closed classes have a ﬁnite number of obstructions.

x ∈ D ⇐⇒ m1 6 x ∧ · · · ∧ mk 6 x

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 3 / 14 Why do we like well-quasi-orders?

Upwards closed classes have a ﬁnite number of minimal elements.

x ∈ U ⇐⇒ m1 6 x ∨ · · · ∨ mk 6 x (ﬁnite base) Membership testing can be done in a ﬁnite number of checks.

m1 m3

m2 Downwards closed classes have a ﬁnite number of obstructions.

x ∈ D ⇐⇒ m1 6 x ∧ · · · ∧ mk 6 x

Membership testing can be done in a ﬁnite number of checks.

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 3 / 14 Why do we like well-quasi-orders?

graphs of genus > g + minor relation (> k)-colorable graphs + induced subgraph relation ...

m1 m3

graphs of treewidth 6 k + minor relation trees + contraction relation ...

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 3 / 14 contraction relation 6ctr: E contraction; induced minor relation 6im: V deletion and E contraction; minor relation 6m: V and E deletion, E contraction.

In this talk...

(S, 6) = (graph class, graph containment relation)

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 4 / 14 In this talk...

(S, 6) = (graph class, graph containment relation)

contraction relation 6ctr: E contraction; induced minor relation 6im: V deletion and E contraction; minor relation 6m: V and E deletion, E contraction.

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 4 / 14 ...some are not: (induced) subgraphs contractions induced minors (induced) topological minor ...and some are still open: induced immersions strong immersions

Main message: decomposition results (sometimes) imply WQO.

Containment relations on graphs

Some are WQO: minors: [GMXX] (Wagner’s conjecture) immersions: [GMXXIII] (Nash-Williams’ Conjecture)

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 5 / 14 ...and some are still open: induced immersions strong immersions

Main message: decomposition results (sometimes) imply WQO.

Containment relations on graphs

Some are WQO: minors: [GMXX] (Wagner’s conjecture) immersions: [GMXXIII] (Nash-Williams’ Conjecture) ...some are not: (induced) subgraphs contractions induced minors (induced) topological minor

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 5 / 14 Main message: decomposition results (sometimes) imply WQO.

Containment relations on graphs

Some are WQO: minors: [GMXX] (Wagner’s conjecture) immersions: [GMXXIII] (Nash-Williams’ Conjecture) ...some are not: (induced) subgraphs contractions induced minors (induced) topological minor ...and some are still open: induced immersions strong immersions

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 5 / 14 Containment relations on graphs

Main message: decomposition results (sometimes) imply WQO.

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 5 / 14 3 show that encodings are WQO by this order; 4 that’s it! antichain {G1, G2,... } ⇒ antichain {enc(G1), enc(G2),... }

1 choose an encoding of graphs as tuples (for instance) e.g. # of subdivisions for each edge, in some chosen order; enc( ) = (1, 0, 1, 0, 2)

0 0 2 choose an order on tuples s.t. enc(G) 6 enc(G ) ⇒ G 6ctr G e.g. the product order, (2, 1, 0, 3, 1) 6 (5, 1, 2, 4, 1)

From structure to WQO

Toy example Subdivisions of H are WQO by contraction.

H =

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 6 / 14 4 that’s it! antichain {G1, G2,... } ⇒ antichain {enc(G1), enc(G2),... }

0 0 2 choose an order on tuples s.t. enc(G) 6 enc(G ) ⇒ G 6ctr G e.g. the product order, (2, 1, 0, 3, 1) 6 (5, 1, 2, 4, 1) 3 show that encodings are WQO by this order;

From structure to WQO

Toy example Subdivisions of H are WQO by contraction. 4 H = 5 3 1 2

1 choose an encoding of graphs as tuples (for instance) e.g. # of subdivisions for each edge, in some chosen order; enc( ) = (1, 0, 1, 0, 2)

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 6 / 14 3 show that encodings are WQO by this order; 4 that’s it! antichain {G1, G2,... } ⇒ antichain {enc(G1), enc(G2),... }

From structure to WQO

Toy example Subdivisions of H are WQO by contraction. 4 H = 5 3 1 2

1 choose an encoding of graphs as tuples (for instance) e.g. # of subdivisions for each edge, in some chosen order; enc( ) = (1, 0, 1, 0, 2)

0 0 2 choose an order on tuples s.t. enc(G) 6 enc(G ) ⇒ G 6ctr G e.g. the product order, (2, 1, 0, 3, 1) 6 (5, 1, 2, 4, 1)

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 6 / 14 4 that’s it! antichain {G1, G2,... } ⇒ antichain {enc(G1), enc(G2),... }

From structure to WQO

Toy example Subdivisions of H are WQO by contraction. 4 H = 5 3 1 2

1 choose an encoding of graphs as tuples (for instance) e.g. # of subdivisions for each edge, in some chosen order; enc( ) = (1, 0, 1, 0, 2)

0 0 2 choose an order on tuples s.t. enc(G) 6 enc(G ) ⇒ G 6ctr G e.g. the product order, (2, 1, 0, 3, 1) 6 (5, 1, 2, 4, 1) 3 show that encodings are WQO by this order;

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 6 / 14 From structure to WQO

Toy example Subdivisions of H are WQO by contraction. 4 H = 5 3 1 2

1 choose an encoding of graphs as tuples (for instance) e.g. # of subdivisions for each edge, in some chosen order; enc( ) = (1, 0, 1, 0, 2)

0 0 2 choose an order on tuples s.t. enc(G) 6 enc(G ) ⇒ G 6ctr G e.g. the product order, (2, 1, 0, 3, 1) 6 (5, 1, 2, 4, 1) 3 show that encodings are WQO by this order; 4 that’s it! antichain {G1, G2,... } ⇒ antichain {enc(G1), enc(G2),... }

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 6 / 14 1 choose a mapping enc : G → S where S is simple; 0 0 2 choose an order S on S s.t. enc(G) S enc(G ) ⇒ G G ;

3 show that (S, S ) is a WQO.

In other words: order-preserving functions send WQOs on WQOs.

Choice of S: given by a decomposition; S = (labeled) 2-c graphs of G; S ⊆ G? (Nash-Williams’ minimum bad sequence).

High level view

Goal: show that (G, 6) is a WQO.

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 7 / 14 0 0 2 choose an order S on S s.t. enc(G) S enc(G ) ⇒ G G ;

3 show that (S, S ) is a WQO.

In other words: order-preserving functions send WQOs on WQOs.

Choice of S: given by a decomposition; S = (labeled) 2-c graphs of G; S ⊆ G? (Nash-Williams’ minimum bad sequence).

High level view

Goal: show that (G, 6) is a WQO. 1 choose a mapping enc : G → S where S is simple;

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 7 / 14 3 show that (S, S ) is a WQO.

In other words: order-preserving functions send WQOs on WQOs.

Choice of S: given by a decomposition; S = (labeled) 2-c graphs of G; S ⊆ G? (Nash-Williams’ minimum bad sequence).

High level view

Goal: show that (G, 6) is a WQO. 1 choose a mapping enc : G → S where S is simple; 0 0 2 choose an order S on S s.t. enc(G) S enc(G ) ⇒ G G ;

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 7 / 14 In other words: order-preserving functions send WQOs on WQOs.

Choice of S: given by a decomposition; S = (labeled) 2-c graphs of G; S ⊆ G? (Nash-Williams’ minimum bad sequence).

High level view

Goal: show that (G, 6) is a WQO. 1 choose a mapping enc : G → S where S is simple; 0 0 2 choose an order S on S s.t. enc(G) S enc(G ) ⇒ G G ;

3 show that (S, S ) is a WQO.

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 7 / 14 Choice of S: given by a decomposition; S = (labeled) 2-c graphs of G; S ⊆ G? (Nash-Williams’ minimum bad sequence).

High level view

Goal: show that (G, 6) is a WQO. 1 choose a mapping enc : G → S where S is simple; 0 0 2 choose an order S on S s.t. enc(G) S enc(G ) ⇒ G G ;

3 show that (S, S ) is a WQO.

In other words: order-preserving functions send WQOs on WQOs.

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 7 / 14 S = (labeled) 2-c graphs of G; S ⊆ G? (Nash-Williams’ minimum bad sequence).

High level view

Goal: show that (G, 6) is a WQO. 1 choose a mapping enc : G → S where S is simple; 0 0 2 choose an order S on S s.t. enc(G) S enc(G ) ⇒ G G ;

3 show that (S, S ) is a WQO.

In other words: order-preserving functions send WQOs on WQOs.

Choice of S: given by a decomposition;

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 7 / 14 S ⊆ G? (Nash-Williams’ minimum bad sequence).

High level view

Goal: show that (G, 6) is a WQO. 1 choose a mapping enc : G → S where S is simple; 0 0 2 choose an order S on S s.t. enc(G) S enc(G ) ⇒ G G ;

3 show that (S, S ) is a WQO.

In other words: order-preserving functions send WQOs on WQOs.

Choice of S: given by a decomposition; S = (labeled) 2-c graphs of G;

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 7 / 14 High level view

Goal: show that (G, 6) is a WQO. 1 choose a mapping enc : G → S where S is simple; 0 0 2 choose an order S on S s.t. enc(G) S enc(G ) ⇒ G G ;

3 show that (S, S ) is a WQO.

In other words: order-preserving functions send WQOs on WQOs.

Choice of S: given by a decomposition; S = (labeled) 2-c graphs of G; S ⊆ G? (Nash-Williams’ minimum bad sequence).

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 7 / 14 Lemma -induced minor-free graphs are WQO by induced minors.

First example

Theorem (Blasiok, Kami´nski,R., Trunck ’15)

H-induced minor-free graphs are WQO by 6im iﬀ H is induced minor of or .

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 8 / 14 First example

Theorem (Blasiok, Kami´nski,R., Trunck ’15)

H-induced minor-free graphs are WQO by 6im iﬀ H is induced minor of or .

Lemma -induced minor-free graphs are WQO by induced minors.

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 8 / 14 enc(G) = (G1, λ1),..., (Gk , λk )

Gi is a path or a cograph; 6 λi : V (Gi ) → {0, 1} (labeling) we choose an order on encodings.

(labeled) cographs and paths are easy to order.

Following the recipe

Lemma (Blasiok, Kami´nski,R., Trunck ’15) Every 2-connected -induced minor-free graph is like this:

C P C: cograph . P: path . C

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 9 / 14 Gi is a path or a cograph; 6 λi : V (Gi ) → {0, 1} (labeling) we choose an order on encodings.

(labeled) cographs and paths are easy to order.

Following the recipe

Lemma (Blasiok, Kami´nski,R., Trunck ’15) Every 2-connected -induced minor-free graph is like this:

C P C: cograph . P: path . C

enc(G) = (G1, λ1),..., (Gk , λk )

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 9 / 14 we choose an order on encodings.

(labeled) cographs and paths are easy to order.

Following the recipe

Lemma (Blasiok, Kami´nski,R., Trunck ’15) Every 2-connected -induced minor-free graph is like this:

C P C: cograph . P: path . C

enc(G) = (G1, λ1),..., (Gk , λk )

Gi is a path or a cograph; 6 λi : V (Gi ) → {0, 1} (labeling)

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 9 / 14 (labeled) cographs and paths are easy to order.

Following the recipe

Lemma (Blasiok, Kami´nski,R., Trunck ’15) Every 2-connected -induced minor-free graph is like this:

C P C: cograph . P: path . C

enc(G) = (G1, λ1),..., (Gk , λk )

Gi is a path or a cograph; 6 λi : V (Gi ) → {0, 1} (labeling) we choose an order on encodings.

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 9 / 14 Following the recipe

Lemma (Blasiok, Kami´nski,R., Trunck ’15) Every 2-connected -induced minor-free graph is like this:

C P C: cograph . P: path . C

enc(G) = (G1, λ1),..., (Gk , λk )

Gi is a path or a cograph; 6 λi : V (Gi ) → {0, 1} (labeling) we choose an order on encodings.

(labeled) cographs and paths are easy to order.

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 9 / 14 Lemma -contraction-free graphs are wqo by contractions.

Second example

Theorem (Blasiok, Kami´nski,R., Trunck ’15+) H-contraction-free graphs are WQO by contractions iﬀ H is a contraction of .

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 10 / 14 Second example

Theorem (Blasiok, Kami´nski,R., Trunck ’15+) H-contraction-free graphs are WQO by contractions iﬀ H is a contraction of .

Lemma -contraction-free graphs are wqo by contractions.

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 10 / 14 The decomposition

Lemma If G is -contraction-free then every block of G is either a clique or an induced cycle.

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 11 / 14 ? If( G1,..., Gp) 6ctr (H1,..., Hq) then cycle(G1,..., Gp) 6ctr cycle(H1,..., Hq) and clique(G1,..., Gp) 6ctr clique(H1,..., Hq)

The encoding

, , , , , = cycle( )

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 12 / 14 The encoding

, , , , , = cycle( )

? If( G1,..., Gp) 6ctr (H1,..., Hq) then cycle(G1,..., Gp) 6ctr cycle(H1,..., Hq) and clique(G1,..., Gp) 6ctr clique(H1,..., Hq)

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 12 / 14 2 B = “parts” of Ai ’s; 3 (B, 6ctr) is a wqo, by minimality of {Ai }i ; ? 4 (B , 6ctr) is a wqo; ? ? 5 (cycle(B ) ∪ clique(B ), 6ctr) is a wqo; 6 {Ai }i is WQO, contradiction.

-contraction-free graphs are WQO by contractions.

Well-quasi-ordering encodings

1 Consider a minimal inﬁnite antichain:

A1, A2,...

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 13 / 14 3 (B, 6ctr) is a wqo, by minimality of {Ai }i ; ? 4 (B , 6ctr) is a wqo; ? ? 5 (cycle(B ) ∪ clique(B ), 6ctr) is a wqo; 6 {Ai }i is WQO, contradiction.

-contraction-free graphs are WQO by contractions.

Well-quasi-ordering encodings

1 Consider a minimal inﬁnite antichain:

A1, A2,...

2 B = “parts” of Ai ’s;

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 13 / 14 ? 4 (B , 6ctr) is a wqo; ? ? 5 (cycle(B ) ∪ clique(B ), 6ctr) is a wqo; 6 {Ai }i is WQO, contradiction.

-contraction-free graphs are WQO by contractions.

Well-quasi-ordering encodings

1 Consider a minimal inﬁnite antichain:

A1, A2,...

2 B = “parts” of Ai ’s; 3 (B, 6ctr) is a wqo, by minimality of {Ai }i ;

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 13 / 14 ? ? 5 (cycle(B ) ∪ clique(B ), 6ctr) is a wqo; 6 {Ai }i is WQO, contradiction.

-contraction-free graphs are WQO by contractions.

Well-quasi-ordering encodings

1 Consider a minimal inﬁnite antichain:

A1, A2,...

2 B = “parts” of Ai ’s; 3 (B, 6ctr) is a wqo, by minimality of {Ai }i ; ? 4 (B , 6ctr) is a wqo;

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 13 / 14 6 {Ai }i is WQO, contradiction.

-contraction-free graphs are WQO by contractions.

Well-quasi-ordering encodings

1 Consider a minimal inﬁnite antichain:

A1, A2,...

2 B = “parts” of Ai ’s; 3 (B, 6ctr) is a wqo, by minimality of {Ai }i ; ? 4 (B , 6ctr) is a wqo; ? ? 5 (cycle(B ) ∪ clique(B ), 6ctr) is a wqo;

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 13 / 14 -contraction-free graphs are WQO by contractions.

Well-quasi-ordering encodings

1 Consider a minimal inﬁnite antichain:

A1, A2,...

2 B = “parts” of Ai ’s; 3 (B, 6ctr) is a wqo, by minimality of {Ai }i ; ? 4 (B , 6ctr) is a wqo; ? ? 5 (cycle(B ) ∪ clique(B ), 6ctr) is a wqo; 6 {Ai }i is WQO, contradiction.

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 13 / 14 Well-quasi-ordering encodings

1 Consider a minimal inﬁnite antichain:

A1, A2,...

2 B = “parts” of Ai ’s; 3 (B, 6ctr) is a wqo, by minimality of {Ai }i ; ? 4 (B , 6ctr) is a wqo; ? ? 5 (cycle(B ) ∪ clique(B ), 6ctr) is a wqo; 6 {Ai }i is WQO, contradiction.

-contraction-free graphs are WQO by contractions.

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 13 / 14 needs a good choice of the (encoding, order) (not always possible); well-quasi-orders are “simple” orders.

Further work: study the limit cases for parameterized classes in non-wqos (H-6-free, sparse classes, bounded parameter, etc.); which classes are wqo by strong immersions? Thank you!

Conclusion

general recipe to go from a decomposition to a wqo;

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 14 / 14 well-quasi-orders are “simple” orders.

Further work: study the limit cases for parameterized classes in non-wqos (H-6-free, sparse classes, bounded parameter, etc.); which classes are wqo by strong immersions? Thank you!

Conclusion

general recipe to go from a decomposition to a wqo; needs a good choice of the (encoding, order) (not always possible);

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 14 / 14 Further work: study the limit cases for parameterized classes in non-wqos (H-6-free, sparse classes, bounded parameter, etc.); which classes are wqo by strong immersions? Thank you!

Conclusion

general recipe to go from a decomposition to a wqo; needs a good choice of the (encoding, order) (not always possible); well-quasi-orders are “simple” orders.

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 14 / 14 Thank you!

Conclusion

general recipe to go from a decomposition to a wqo; needs a good choice of the (encoding, order) (not always possible); well-quasi-orders are “simple” orders.

Further work: study the limit cases for parameterized classes in non-wqos (H-6-free, sparse classes, bounded parameter, etc.); which classes are wqo by strong immersions?

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 14 / 14 Conclusion

general recipe to go from a decomposition to a wqo; needs a good choice of the (encoding, order) (not always possible); well-quasi-orders are “simple” orders.

Further work: study the limit cases for parameterized classes in non-wqos (H-6-free, sparse classes, bounded parameter, etc.); which classes are wqo by strong immersions? Thank you!

Jean-Florent Raymond (Montpellier & Warsaw) Graph decompositions and well-quasi-ordering 14 / 14