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Enumerating minimal dominating setsin triangle-free graphs
Oscar Defrain
LIMOS, Clermont-Ferrand.
STACS 2019
March 13–16, TU Berlin.
Joint work with Marthe Bonamy (LaBRI, Bordeaux), Marc Heinrich
(LIRIS, Lyon) and Jean-Florent Raymond (TU Berlin).
Slides of Jean-Florent Raymond.
Introduction to enumeration algorithms
Typical question:
given input I , list all objects of type X in I
• cycles, cliques, stable sets, dominating sets, etc. of a graph
• transversals of a hypergraph
• variable assignments satisfying a formula
• trains to Paris leaving tomorrow before 10:00
• shortest paths from Montpellier to Marseille
• . . .
Remark: possibly many objects!
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 2 / 13
Introduction to enumeration algorithms
Typical question:
given input I , list all objects of type X in I
• cycles, cliques, stable sets, dominating sets, etc. of a graph
• transversals of a hypergraph
• variable assignments satisfying a formula
• trains to Paris leaving tomorrow before 10:00
• shortest paths from Montpellier to Marseille
• . . .
Remark: possibly many objects!
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 2 / 13
Introduction to enumeration algorithms
Typical question:
given input I , list all objects of type X in I
• cycles, cliques, stable sets, dominating sets, etc. of a graph
• transversals of a hypergraph
• variable assignments satisfying a formula
• trains to Paris leaving tomorrow before 10:00
• shortest paths from Montpellier to Marseille
• . . .
Remark: possibly many objects!
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 2 / 13
Introduction to enumeration algorithms
Typical question:
given input I , list all objects of type X in I
• cycles, cliques, stable sets, dominating sets, etc. of a graph
• transversals of a hypergraph
• variable assignments satisfying a formula
• trains to Paris leaving tomorrow before 10:00
• shortest paths from Montpellier to Marseille
• . . .
Remark: possibly many objects!
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 2 / 13
Introduction to enumeration algorithms
Typical question:
given input I , list all objects of type X in I
• cycles, cliques, stable sets, dominating sets, etc. of a graph
• transversals of a hypergraph
• variable assignments satisfying a formula
• trains to Paris leaving tomorrow before 10:00
• shortest paths from Montpellier to Marseille
• . . .
Remark: possibly many objects!
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 2 / 13
Introduction to enumeration algorithms
Typical question:
given input I , list all objects of type X in I
• cycles, cliques, stable sets, dominating sets, etc. of a graph
• transversals of a hypergraph
• variable assignments satisfying a formula
• trains to Paris leaving tomorrow before 10:00
• shortest paths from Montpellier to Marseille
• . . .
Remark: possibly many objects!
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 2 / 13
Introduction to enumeration algorithms
Typical question:
given input I , list all objects of type X in I
• cycles, cliques, stable sets, dominating sets, etc. of a graph
• transversals of a hypergraph
• variable assignments satisfying a formula
• trains to Paris leaving tomorrow before 10:00
• shortest paths from Montpellier to Marseille
• . . .
Remark: possibly many objects!
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 2 / 13
Two perspectives about complexity
Input-sensitive: in terms of input size
• [Fomin, Grandoni, and Pyatkin, 2008]:
O (1.7159n)-time for Minimal Dominating Set
n = |V (G )|• upper-bounds the number of objects
Output-sensitive: in terms of input+output size
• [Fredman and Khachiyan, 1996]:
O(N logN
)-time for Minimal Dominating Set
N = |G |+ |D(G )|
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 3 / 13
Two perspectives about complexity
Input-sensitive: in terms of input size
• [Fomin, Grandoni, and Pyatkin, 2008]:
O (1.7159n)-time for Minimal Dominating Set
n = |V (G )|
• upper-bounds the number of objects
Output-sensitive: in terms of input+output size
• [Fredman and Khachiyan, 1996]:
O(N logN
)-time for Minimal Dominating Set
N = |G |+ |D(G )|
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 3 / 13
Two perspectives about complexity
Input-sensitive: in terms of input size
• [Fomin, Grandoni, and Pyatkin, 2008]:
O (1.7159n)-time for Minimal Dominating Set
n = |V (G )|• upper-bounds the number of objects
Output-sensitive: in terms of input+output size
• [Fredman and Khachiyan, 1996]:
O(N logN
)-time for Minimal Dominating Set
N = |G |+ |D(G )|
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 3 / 13
Two perspectives about complexity
Input-sensitive: in terms of input size
• [Fomin, Grandoni, and Pyatkin, 2008]:
O (1.7159n)-time for Minimal Dominating Set
n = |V (G )|• upper-bounds the number of objects
Output-sensitive: in terms of input+output size
• [Fredman and Khachiyan, 1996]:
O(N logN
)-time for Minimal Dominating Set
N = |G |+ |D(G )|
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 3 / 13
Two perspectives about complexity
Input-sensitive: in terms of input size
• [Fomin, Grandoni, and Pyatkin, 2008]:
O (1.7159n)-time for Minimal Dominating Set
n = |V (G )|• upper-bounds the number of objects
Output-sensitive: in terms of input+output size
• [Fredman and Khachiyan, 1996]:
O(N logN
)-time for Minimal Dominating Set
N = |G |+ |D(G )|
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 3 / 13
“Fast” output-sensitive algorithms
n: input size
s: output size
• output-polynomial: poly(n + s) to get all the solutions
• polynomial delay: poly(n) per solution
Note: output space is not counted.
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 4 / 13
“Fast” output-sensitive algorithms
n: input size
s: output size
• output-polynomial: poly(n + s) to get all the solutions
• polynomial delay: poly(n) per solution
Note: output space is not counted.
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 4 / 13
“Fast” output-sensitive algorithms
n: input size
s: output size
• output-polynomial: poly(n + s) to get all the solutions
• polynomial delay: poly(n) per solution
Note: output space is not counted.
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 4 / 13
Minimal dominating sets
• N(v): neighbors of a vertex v
• dominating set (DS): D ⊆ V (G ) s.t. V (G ) = D ∪ N(D)
“D can see everybody else”
• minimal dominating set: inclusion-wise minimal DS
• private neighbor of v ∈ D:
vertex that is
{dominated by v, and
not dominated by D \ {v}(possibly v)
• irredundant set: S ⊆ V (G ) s.t. every x ∈ S has a priv. neigh.
• set of minimal DS denoted by D(G )
Simple observation:
a DS is minimal if and only if it is irredundant
(if all its vertices have a private neighbor)
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 5 / 13
Minimal dominating sets
• N(v): neighbors of a vertex v
• dominating set (DS): D ⊆ V (G ) s.t. V (G ) = D ∪ N(D)
“D can see everybody else”
• minimal dominating set: inclusion-wise minimal DS
• private neighbor of v ∈ D:
vertex that is
{dominated by v, and
not dominated by D \ {v}(possibly v)
• irredundant set: S ⊆ V (G ) s.t. every x ∈ S has a priv. neigh.
• set of minimal DS denoted by D(G )
Simple observation:
a DS is minimal if and only if it is irredundant
(if all its vertices have a private neighbor)
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 5 / 13
Minimal dominating sets
• N(v): neighbors of a vertex v
• dominating set (DS): D ⊆ V (G ) s.t. V (G ) = D ∪ N(D)
“D can see everybody else”
• minimal dominating set: inclusion-wise minimal DS
• private neighbor of v ∈ D:
vertex that is
{dominated by v, and
not dominated by D \ {v}(possibly v)
• irredundant set: S ⊆ V (G ) s.t. every x ∈ S has a priv. neigh.
• set of minimal DS denoted by D(G )
Simple observation:
a DS is minimal if and only if it is irredundant
(if all its vertices have a private neighbor)
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 5 / 13
Minimal dominating sets
• N(v): neighbors of a vertex v
• dominating set (DS): D ⊆ V (G ) s.t. V (G ) = D ∪ N(D)
“D can see everybody else”
• minimal dominating set: inclusion-wise minimal DS
• private neighbor of v ∈ D:
vertex that is
{dominated by v, and
not dominated by D \ {v}(possibly v)
• irredundant set: S ⊆ V (G ) s.t. every x ∈ S has a priv. neigh.
• set of minimal DS denoted by D(G )
Simple observation:
a DS is minimal if and only if it is irredundant
(if all its vertices have a private neighbor)
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 5 / 13
Minimal dominating sets
• N(v): neighbors of a vertex v
• dominating set (DS): D ⊆ V (G ) s.t. V (G ) = D ∪ N(D)
“D can see everybody else”
• minimal dominating set: inclusion-wise minimal DS
• private neighbor of v ∈ D:
vertex that is
{dominated by v, and
not dominated by D \ {v}(possibly v)
• irredundant set: S ⊆ V (G ) s.t. every x ∈ S has a priv. neigh.
• set of minimal DS denoted by D(G )
Simple observation:
a DS is minimal if and only if it is irredundant
(if all its vertices have a private neighbor)
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 5 / 13
Minimal dominating sets
• N(v): neighbors of a vertex v
• dominating set (DS): D ⊆ V (G ) s.t. V (G ) = D ∪ N(D)
“D can see everybody else”
• minimal dominating set: inclusion-wise minimal DS
• private neighbor of v ∈ D:
vertex that is
{dominated by v, and
not dominated by D \ {v}(possibly v)
• irredundant set: S ⊆ V (G ) s.t. every x ∈ S has a priv. neigh.
• set of minimal DS denoted by D(G )
Simple observation:
a DS is minimal if and only if it is irredundant
(if all its vertices have a private neighbor)
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 5 / 13
Minimal dominating sets
• N(v): neighbors of a vertex v
• dominating set (DS): D ⊆ V (G ) s.t. V (G ) = D ∪ N(D)
“D can see everybody else”
• minimal dominating set: inclusion-wise minimal DS
• private neighbor of v ∈ D:
vertex that is
{dominated by v, and
not dominated by D \ {v}(possibly v)
• irredundant set: S ⊆ V (G ) s.t. every x ∈ S has a priv. neigh.
• set of minimal DS denoted by D(G )
Simple observation:
a DS is minimal if and only if it is irredundant
(if all its vertices have a private neighbor)
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 5 / 13
Enumeration of minimal dominating sets
Problem: enumerating D(G ) given G .
Dream goal: an output-polynomial time algorithm
Running time polynomial in (input size + output size)
General case: open (best is quasi-polynomial)
Known: degenerate graphs, graphs of bounded clique-width or
LMIM-width, split graphs, chordal graph, graphs of girth > 7, etc.
Theorem (Bonamy, D., Heinrich, Raymond, 2018+)
There is an output-polynomial time algorithm enumerating
minimal dominating sets in triangle-free graphs.
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 6 / 13
Enumeration of minimal dominating sets
Problem: enumerating D(G ) given G .
Dream goal: an output-polynomial time algorithm
Running time polynomial in (input size + output size)
General case: open (best is quasi-polynomial)
Known: degenerate graphs, graphs of bounded clique-width or
LMIM-width, split graphs, chordal graph, graphs of girth > 7, etc.
Theorem (Bonamy, D., Heinrich, Raymond, 2018+)
There is an output-polynomial time algorithm enumerating
minimal dominating sets in triangle-free graphs.
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 6 / 13
Enumeration of minimal dominating sets
Problem: enumerating D(G ) given G .
Dream goal: an output-polynomial time algorithm
Running time polynomial in (input size + output size)
General case: open (best is quasi-polynomial)
Known: degenerate graphs, graphs of bounded clique-width or
LMIM-width, split graphs, chordal graph, graphs of girth > 7, etc.
Theorem (Bonamy, D., Heinrich, Raymond, 2018+)
There is an output-polynomial time algorithm enumerating
minimal dominating sets in triangle-free graphs.
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 6 / 13
Enumeration of minimal dominating sets
Problem: enumerating D(G ) given G .
Dream goal: an output-polynomial time algorithm
Running time polynomial in (input size + output size)
General case: open (best is quasi-polynomial)
Known: degenerate graphs, graphs of bounded clique-width or
LMIM-width, split graphs, chordal graph, graphs of girth > 7, etc.
Theorem (Bonamy, D., Heinrich, Raymond, 2018+)
There is an output-polynomial time algorithm enumerating
minimal dominating sets in triangle-free graphs.
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 6 / 13
Minimal dominating sets in split graphs [KLMN14]
C S
stable set (maximal)clique
D minimal DS in G
mD dominates S and every v ∈ D ∩ C has a priv. neigh. in S
Enumeration: complete every irredundant X ⊆ C in S
linear delay [Kante, Limouzy, Mary, and Nourine, 2014]
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 7 / 13
Minimal dominating sets in split graphs [KLMN14]
C S
stable set (maximal)clique
D minimal DS in G
mD dominates S and every v ∈ D ∩ C has a priv. neigh. in S
Enumeration: complete every irredundant X ⊆ C in S
linear delay [Kante, Limouzy, Mary, and Nourine, 2014]
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 7 / 13
Minimal dominating sets in split graphs [KLMN14]
C S
stable set (maximal)clique
D minimal DS in G
mD dominates S and every v ∈ D ∩ C has a priv. neigh. in S
Enumeration: complete every irredundant X ⊆ C in S
linear delay [Kante, Limouzy, Mary, and Nourine, 2014]
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 7 / 13
Minimal dominating sets in split graphs [KLMN14]
C S
stable set (maximal)clique
D minimal DS in G
mD dominates S and every v ∈ D ∩ C has a priv. neigh. in S
Then D ∩ S = {all vertices not dominated by D ∩ C}
Enumeration: complete every irredundant X ⊆ C in S
linear delay [Kante, Limouzy, Mary, and Nourine, 2014]
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 7 / 13
Minimal dominating sets in split graphs [KLMN14]
C S
stable set (maximal)clique
D minimal DS in G
mD dominates S and every v ∈ D ∩ C has a priv. neigh. in S
Then D ∩ S = {all vertices not dominated by D ∩ C}
Enumeration: complete every irredundant X ⊆ C in S
linear delay [Kante, Limouzy, Mary, and Nourine, 2014]
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 7 / 13
Minimal dominating sets in split graphs [KLMN14]
C S
stable set (maximal)clique
D minimal DS in G
mD dominates S and every v ∈ D ∩ C has a priv. neigh. in S
Then D ∩ S = {all vertices not dominated by D ∩ C}
Enumeration: complete every irredundant X ⊆ C in S
linear delay [Kante, Limouzy, Mary, and Nourine, 2014]
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 7 / 13
Minimal dominating sets in split graphs [KLMN14]
C S
stable set (maximal)clique
D minimal DS in G
mD dominates S and every v ∈ D ∩ C has a priv. neigh. in S
Then D ∩ S = {all vertices not dominated by D ∩ C}
Enumeration: complete every irredundant X ⊆ C in S
linear delay [Kante, Limouzy, Mary, and Nourine, 2014]
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 7 / 13
Peeling graphs
Goal: enumeration of minimal DS in triangle-free graphs
G
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 8 / 13
Peeling graphs
Goal: enumeration of minimal DS in triangle-free graphs
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 8 / 13
Peeling graphs
Goal: enumeration of minimal DS in triangle-free graphs
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 8 / 13
Peeling graphs
Goal: enumeration of minimal DS in triangle-free graphs
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 8 / 13
Peeling graphs
Goal: enumeration of minimal DS in triangle-free graphs
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 8 / 13
Peeling graphs
Goal: enumeration of minimal DS in triangle-free graphs
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 8 / 13
Peeling graphs
Goal: enumeration of minimal DS in triangle-free graphs
...
∅
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 8 / 13
Peeling graphs
Goal: enumeration of minimal DS in triangle-free graphs
...
∅
up
ui
u1
...
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 8 / 13
Peeling graphs
Goal: enumeration of minimal DS in triangle-free graphs
...
∅
up
ui
u1
...
V0
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 8 / 13
Peeling graphs
Goal: enumeration of minimal DS in triangle-free graphs
...
∅
up
ui
u1
...
V1
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 8 / 13
Peeling graphs
Goal: enumeration of minimal DS in triangle-free graphs
...
∅
up
ui
u1
...
Vi
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 8 / 13
Peeling graphs
Goal: enumeration of minimal DS in triangle-free graphs
...
∅
up
ui
u1
...
Vi
Plan:
1. given minimal DS of Vi
allowing vertices of G − Vi
2. enumerate those of Vi+1
allowing vertices of G − Vi+1
extend each minimal DS of Vi
to minimal DS of Vi+1
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 8 / 13
Peeling graphs
Goal: enumeration of minimal DS in triangle-free graphs
...
∅
up
ui
u1
...
Vi
Plan:
1. given minimal DS of Vi
allowing vertices of G − Vi
2. enumerate those of Vi+1
allowing vertices of G − Vi+1
extend each minimal DS of Vi
to minimal DS of Vi+1
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 8 / 13
Peeling graphs
Goal: enumeration of minimal DS in triangle-free graphs
...
∅
up
ui
u1
...
Vi
Plan:
1. given minimal DS of Vi
allowing vertices of G − Vi
2. enumerate those of Vi+1
allowing vertices of G − Vi+1
extend each minimal DS of Vi
to minimal DS of Vi+1
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 8 / 13
Peeling graphs
Goal: enumeration of minimal DS in triangle-free graphs
...
∅
up
ui
u1
...
Vi
Plan:
1. given minimal DS of Vi
allowing vertices of G − Vi
2. enumerate those of Vi+1
allowing vertices of G − Vi+1
extend each minimal DS of Vi
to minimal DS of Vi+1
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 8 / 13
Growing partial minimal dominating sets
V0
V1
V2
...
Vp
∅
Important wanted properties:
• no cycle (using a parent relation: lexicographical order)
• no leaf before level p (no exponential blowup)
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 9 / 13
Computing Vi+1 from Vi
Goal: extend a minimal DS D of Vi to a minimal DS of Vi+1
ui+1
Vi
Observation:
• possibly D dominates Vi+1
• if not then D ∪ {ui+1} does.
candidate extension: minimal X s.t. D ∪ X dominates Vi+1
Lemma: |candidate extensions of D| 6 |minimal DS of G |(so we can try them all even if only few work)
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 10 / 13
Computing Vi+1 from Vi
Goal: extend a minimal DS D of Vi to a minimal DS of Vi+1
ui+1
Vi
Observation:
• possibly D dominates Vi+1
• if not then D ∪ {ui+1} does.
candidate extension: minimal X s.t. D ∪ X dominates Vi+1
Lemma: |candidate extensions of D| 6 |minimal DS of G |(so we can try them all even if only few work)
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 10 / 13
Computing Vi+1 from Vi
Goal: extend a minimal DS D of Vi to a minimal DS of Vi+1
ui+1
Vi
Observation:
• possibly D dominates Vi+1
• if not then D ∪ {ui+1} does.
candidate extension: minimal X s.t. D ∪ X dominates Vi+1
Lemma: |candidate extensions of D| 6 |minimal DS of G |(so we can try them all even if only few work)
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 10 / 13
Computing Vi+1 from Vi
Goal: extend a minimal DS D of Vi to a minimal DS of Vi+1
ui+1
Vi
Observation:
• possibly D dominates Vi+1
• if not then D ∪ {ui+1} does.
extension is always possible, hence
|minimal DS of Vi | 6 |minimal DS of Vi+1|6 |minimal DS of G |
candidate extension: minimal X s.t. D ∪ X dominates Vi+1
Lemma: |candidate extensions of D| 6 |minimal DS of G |(so we can try them all even if only few work)
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 10 / 13
Computing Vi+1 from Vi
Goal: extend a minimal DS D of Vi to a minimal DS of Vi+1
ui+1
Vi
Observation:
• possibly D dominates Vi+1
• if not then D ∪ {ui+1} does.
extension is always possible, hence
|minimal DS of Vi | 6 |minimal DS of Vi+1|6 |minimal DS of G |
candidate extension: minimal X s.t. D ∪ X dominates Vi+1
Lemma: |candidate extensions of D| 6 |minimal DS of G |(so we can try them all even if only few work)
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 10 / 13
Computing Vi+1 from Vi
Goal: extend a minimal DS D of Vi to a minimal DS of Vi+1
ui+1
Vi
Observation:
• possibly D dominates Vi+1
• if not then D ∪ {ui+1} does.
extension is always possible, hence
|minimal DS of Vi | 6 |minimal DS of Vi+1|6 |minimal DS of G |
candidate extension: minimal X s.t. D ∪ X dominates Vi+1
Lemma: |candidate extensions of D| 6 |minimal DS of G |(so we can try them all even if only few work)
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 10 / 13
Which are the candidate extensions of D?
D: minimal DS of Vi
Candidate ext.: minimal X s.t. D ∪ X dominates Vi+1
Vi
ui+1
• if ui+1 dom. by D: they only have to dominate S
→ exactly the minimal DS of Split(C ,S)
• or ui+1 not dom. by D: they should also dom. ui+1
→ irredundant {t} ∪ Q s.t.
t ∈ N(ui+1)
Q ⊆ C minimal DS of Split(C ,S)
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 11 / 13
Which are the candidate extensions of D?
D: minimal DS of Vi
Candidate ext.: minimal X s.t. D ∪ X dominates Vi+1
Vi
ui+1
N(ui+1) ∩ Vi+1 stable set!
• if ui+1 dom. by D: they only have to dominate S
→ exactly the minimal DS of Split(C ,S)
• or ui+1 not dom. by D: they should also dom. ui+1
→ irredundant {t} ∪ Q s.t.
t ∈ N(ui+1)
Q ⊆ C minimal DS of Split(C ,S)
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 11 / 13
Which are the candidate extensions of D?
D: minimal DS of Vi
Candidate ext.: minimal X s.t. D ∪ X dominates Vi+1
Vi
ui+1
N(ui+1) ∩ Vi+1 stable set!
dominated by D
• if ui+1 dom. by D: they only have to dominate S
→ exactly the minimal DS of Split(C ,S)
• or ui+1 not dom. by D: they should also dom. ui+1
→ irredundant {t} ∪ Q s.t.
t ∈ N(ui+1)
Q ⊆ C minimal DS of Split(C ,S)
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 11 / 13
Which are the candidate extensions of D?
D: minimal DS of Vi
Candidate ext.: minimal X s.t. D ∪ X dominates Vi+1
Vi
ui+1
N(ui+1) ∩ Vi+1 stable set!
dominated by D
SCN(S) \ {ui+1}
• if ui+1 dom. by D: they only have to dominate S
→ exactly the minimal DS of Split(C ,S)
• or ui+1 not dom. by D: they should also dom. ui+1
→ irredundant {t} ∪ Q s.t.
t ∈ N(ui+1)
Q ⊆ C minimal DS of Split(C ,S)
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 11 / 13
Which are the candidate extensions of D?
D: minimal DS of Vi
Candidate ext.: minimal X s.t. D ∪ X dominates Vi+1
Vi
ui+1
N(ui+1) ∩ Vi+1 stable set!
dominated by D
SCN(S) \ {ui+1}
• if ui+1 dom. by D: they only have to dominate S
→ exactly the minimal DS of Split(C ,S)
• or ui+1 not dom. by D: they should also dom. ui+1
→ irredundant {t} ∪ Q s.t.
t ∈ N(ui+1)
Q ⊆ C minimal DS of Split(C ,S)
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 11 / 13
Which are the candidate extensions of D?
D: minimal DS of Vi
Candidate ext.: minimal X s.t. D ∪ X dominates Vi+1
Vi
ui+1
N(ui+1) ∩ Vi+1 stable set!
dominated by D
SCN(S) \ {ui+1}
• if ui+1 dom. by D: they only have to dominate S
→ exactly the minimal DS of Split(C , S)
• or ui+1 not dom. by D: they should also dom. ui+1
→ irredundant {t} ∪ Q s.t.
t ∈ N(ui+1)
Q ⊆ C minimal DS of Split(C ,S)
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 11 / 13
Which are the candidate extensions of D?
D: minimal DS of Vi
Candidate ext.: minimal X s.t. D ∪ X dominates Vi+1
Vi
ui+1
N(ui+1) ∩ Vi+1 stable set!
dominated by D
SCN(S) \ {ui+1}
• if ui+1 dom. by D: they only have to dominate S
→ exactly the minimal DS of Split(C , S)
• or ui+1 not dom. by D: they should also dom. ui+1
→ irredundant {t} ∪ Q s.t.
t ∈ N(ui+1)
Q ⊆ C minimal DS of Split(C ,S)
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 11 / 13
Which are the candidate extensions of D?
D: minimal DS of Vi
Candidate ext.: minimal X s.t. D ∪ X dominates Vi+1
Vi
ui+1
N(ui+1) ∩ Vi+1 stable set!
dominated by D
SCN(S) \ {ui+1}
• if ui+1 dom. by D: they only have to dominate S
→ exactly the minimal DS of Split(C , S)
• or ui+1 not dom. by D: they should also dom. ui+1
→ irredundant {t} ∪ Q s.t.
t ∈ N(ui+1)
Q ⊆ C minimal DS of Split(C ,S)
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 11 / 13
The algorithm
V0
V1
V2
...
Vp
∅
For each D minimal DS of Vi :
• compute all candidate extensions; in time O(poly(n) · |D(G )|)• only keep the X ∪ D’s that are minimal and children of D.
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 12 / 13
The algorithm
V0
V1
V2
...
Vp
∅
For each D minimal DS of Vi :
• compute all candidate extensions; in time O(poly(n) · |D(G )|)• only keep the X ∪ D’s that are minimal and children of D.
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 12 / 13
The algorithm
V0
V1
V2
...
Vp
∅
For each D minimal DS of Vi :
• compute all candidate extensions;
in time O(poly(n) · |D(G )|)• only keep the X ∪ D’s that are minimal and children of D.
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 12 / 13
The algorithm
V0
V1
V2
...
Vp
∅
For each D minimal DS of Vi :
• compute all candidate extensions; in time O(poly(n) · |D(G )|)
• only keep the X ∪ D’s that are minimal and children of D.
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 12 / 13
The algorithm
V0
V1
V2
...
Vp
∅
For each D minimal DS of Vi :
• compute all candidate extensions; in time O(poly(n) · |D(G )|)• only keep the X ∪ D’s that are minimal and children of D.
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 12 / 13
Results and problems
Theorem (Bonamy, D., Heinrich, Raymond, 2018+)
The set D(G ) of minimal dominating sets of any triangle-free
graph G can be enumerated in time poly(n) · |D(G )|2 and
polynomial space.
Theorem (Bonamy, D., Heinrich, Raymond, 2018+)
Deciding if a vertex set X can be extended into a minimal
dominating set is NP-complete in bipartite graphs.
• complexity improvements?
• extensions to other classes? Kt-free and variants
Thank you!
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 13 / 13
Results and problems
Theorem (Bonamy, D., Heinrich, Raymond, 2018+)
The set D(G ) of minimal dominating sets of any triangle-free
graph G can be enumerated in time poly(n) · |D(G )|2 and
polynomial space.
Theorem (Bonamy, D., Heinrich, Raymond, 2018+)
Deciding if a vertex set X can be extended into a minimal
dominating set is NP-complete in bipartite graphs.
• complexity improvements?
• extensions to other classes? Kt-free and variants
Thank you!
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 13 / 13
Results and problems
Theorem (Bonamy, D., Heinrich, Raymond, 2018+)
The set D(G ) of minimal dominating sets of any triangle-free
graph G can be enumerated in time poly(n) · |D(G )|2 and
polynomial space.
Theorem (Bonamy, D., Heinrich, Raymond, 2018+)
Deciding if a vertex set X can be extended into a minimal
dominating set is NP-complete in bipartite graphs.
• complexity improvements?
• extensions to other classes? Kt-free and variants
Thank you!
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 13 / 13
Results and problems
Theorem (Bonamy, D., Heinrich, Raymond, 2018+)
The set D(G ) of minimal dominating sets of any triangle-free
graph G can be enumerated in time poly(n) · |D(G )|2 and
polynomial space.
Theorem (Bonamy, D., Heinrich, Raymond, 2018+)
Deciding if a vertex set X can be extended into a minimal
dominating set is NP-complete in bipartite graphs.
• complexity improvements?
• extensions to other classes?
Kt-free and variants
Thank you!
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 13 / 13
Results and problems
Theorem (Bonamy, D., Heinrich, Raymond, 2018+)
The set D(G ) of minimal dominating sets of any triangle-free
graph G can be enumerated in time poly(n) · |D(G )|2 and
polynomial space.
Theorem (Bonamy, D., Heinrich, Raymond, 2018+)
Deciding if a vertex set X can be extended into a minimal
dominating set is NP-complete in bipartite graphs.
• complexity improvements?
• extensions to other classes? Kt-free and variants
Thank you!
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 13 / 13
Results and problems
Theorem (Bonamy, D., Heinrich, Raymond, 2018+)
The set D(G ) of minimal dominating sets of any triangle-free
graph G can be enumerated in time poly(n) · |D(G )|2 and
polynomial space.
Theorem (Bonamy, D., Heinrich, Raymond, 2018+)
Deciding if a vertex set X can be extended into a minimal
dominating set is NP-complete in bipartite graphs.
• complexity improvements?
• extensions to other classes? Kt-free and variants
Thank you!
Oscar Defrain Enumerating minimal dominating sets in triangle-free graphs 13 / 13