Graph decompositions and well-quasi-ordering
Jean-Florent Raymond
LIRMM, University of Montpellier, France, and
MIMUW, University of Warsaw, Poland
Orléans, November 2015
Jean-Florent Raymond (Montpellier & Warsaw)
Graph decompositions and well-quasi-ordering
1 / 14
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
Well-quasi-ordering
A WQO is an order where:
every decreasing sequence is finite;
every sequence of non-comparable elements is finite.
Examples:
1, 12 , 13 , . . . 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
Well-quasi-ordering
A WQO is an order where:
every decreasing sequence is finite;
every sequence of non-comparable elements is finite.
Examples:
1, 12 , 13 , . . . 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, 12 , 13 , . . . 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
m2
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
m2
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
m2
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
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
b
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 66 x ∧ · · · ∧ mk 66 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 66 x ∧ · · · ∧ mk 66 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
m2
graphs of treewidth 6 k + minor relation
trees + contraction relation
...
Jean-Florent Raymond (Montpellier & Warsaw)
Graph decompositions and well-quasi-ordering
3 / 14
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
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
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
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
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
Main message: decomposition results (sometimes) imply WQO.
Jean-Florent Raymond (Montpellier & Warsaw)
Graph decompositions and well-quasi-ordering
5 / 14
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
From structure to WQO
Toy example
Subdivisions of H are WQO by contraction.
4
5 3 1
H=
2
1
choose an encoding of graphs as tuples (for instance)
e.g. # of subdivisions for each edge, in some chosen order;
enc(
Jean-Florent Raymond (Montpellier & Warsaw)
) = (1, 0, 1, 0, 2)
Graph decompositions and well-quasi-ordering
6 / 14
From structure to WQO
Toy example
Subdivisions of H are WQO by contraction.
4
5 3 1
H=
2
1
choose an encoding of graphs as tuples (for instance)
e.g. # of subdivisions for each edge, in some chosen order;
enc(
2
) = (1, 0, 1, 0, 2)
choose an order on tuples s.t. enc(G ) 6 enc(G 0 ) ⇒ G 6ctr G 0
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
From structure to WQO
Toy example
Subdivisions of H are WQO by contraction.
4
5 3 1
H=
2
1
choose an encoding of graphs as tuples (for instance)
e.g. # of subdivisions for each edge, in some chosen order;
enc(
2
3
) = (1, 0, 1, 0, 2)
choose an order on tuples s.t. enc(G ) 6 enc(G 0 ) ⇒ G 6ctr G 0
e.g. the product order, (2, 1, 0, 3, 1) 6 (5, 1, 2, 4, 1)
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
5 3 1
H=
2
1
choose an encoding of graphs as tuples (for instance)
e.g. # of subdivisions for each edge, in some chosen order;
enc(
2
) = (1, 0, 1, 0, 2)
choose an order on tuples s.t. enc(G ) 6 enc(G 0 ) ⇒ G 6ctr G 0
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
High level view
Goal: show that (G, 6) is a WQO.
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;
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;
2
choose an order S on S s.t. enc(G ) S enc(G 0 ) ⇒ G G 0 ;
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;
2
choose an order S on S s.t. enc(G ) S enc(G 0 ) ⇒ G G 0 ;
3
show that (S, S ) is a WQO.
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;
2
choose an order S on S s.t. enc(G ) S enc(G 0 ) ⇒ G G 0 ;
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
High level view
Goal: show that (G, 6) is a WQO.
1
choose a mapping enc : G → S where S is simple;
2
choose an order S on S s.t. enc(G ) S enc(G 0 ) ⇒ G G 0 ;
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
High level view
Goal: show that (G, 6) is a WQO.
1
choose a mapping enc : G → S where S is simple;
2
choose an order S on S s.t. enc(G ) S enc(G 0 ) ⇒ G G 0 ;
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;
2
choose an order S on S s.t. enc(G ) S enc(G 0 ) ⇒ G G 0 ;
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
First example
Theorem (Blasiok, Kamiński, R., Trunck ’15)
H-induced minor-free graphs are WQO by 6im iff H is induced
or
.
minor of
Jean-Florent Raymond (Montpellier & Warsaw)
Graph decompositions and well-quasi-ordering
8 / 14
First example
Theorem (Blasiok, Kamiński, R., Trunck ’15)
H-induced minor-free graphs are WQO by 6im iff H is induced
or
.
minor of
Lemma
-induced minor-free graphs are WQO by induced minors.
Jean-Florent Raymond (Montpellier & Warsaw)
Graph decompositions and well-quasi-ordering
8 / 14
Following the recipe
Lemma (Blasiok, Kamiński, R., Trunck ’15)
Every 2-connected
-induced minor-free graph is like this:
C
P
..
.
C
Jean-Florent Raymond (Montpellier & Warsaw)
C : cograph
P: path
Graph decompositions and well-quasi-ordering
9 / 14
Following the recipe
Lemma (Blasiok, Kamiński, R., Trunck ’15)
Every 2-connected
-induced minor-free graph is like this:
C
P
..
.
C
C : cograph
P: path
enc(G ) = (G1 , λ1 ), . . . , (Gk , λk )
Jean-Florent Raymond (Montpellier & Warsaw)
Graph decompositions and well-quasi-ordering
9 / 14
Following the recipe
Lemma (Blasiok, Kamiński, R., Trunck ’15)
Every 2-connected
-induced minor-free graph is like this:
C
P
..
.
C
C : cograph
P: path
enc(G ) = (G1 , λ1 ), . . . , (Gk , λk )
Gi is a path or a cograph;
λi : V (Gi ) → {0, 1}6 (labeling)
Jean-Florent Raymond (Montpellier & Warsaw)
Graph decompositions and well-quasi-ordering
9 / 14
Following the recipe
Lemma (Blasiok, Kamiński, R., Trunck ’15)
Every 2-connected
-induced minor-free graph is like this:
C
P
..
.
C
C : cograph
P: path
enc(G ) = (G1 , λ1 ), . . . , (Gk , λk )
Gi is a path or a cograph;
λi : V (Gi ) → {0, 1}6 (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, Kamiński, R., Trunck ’15)
Every 2-connected
-induced minor-free graph is like this:
C
P
..
.
C
C : cograph
P: path
enc(G ) = (G1 , λ1 ), . . . , (Gk , λk )
Gi is a path or a cograph;
λi : V (Gi ) → {0, 1}6 (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
Second example
Theorem (Blasiok, Kamiński, 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, Kamiński, 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
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
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
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
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
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
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
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
Conclusion
general recipe to go from a decomposition to a wqo;
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);
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.
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?
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