Power-domination in triangulations
Claire Pennarun
Joint work with Paul Dorbec and Antonio Gonzalez
LaBRI, Université de Bordeaux
Universidad de Cadiz
JGA, Orléans, 6 novembre 2015
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
1/9
Power domination
Control the world a system with a minimal number of captors [Baldwin
et al. ’91, ’93]
Some vertices in a starting set S (captors).
N [ S] = M
(propagation step) u ∈ M. If v ∈ N (u) is the only vertex outside
N [u] ∪ M: M → M ∪ {x}.
S is a power dominating set (PDS) if M = V (G) at the end.
γP (G) (power domination number of G): minimum size of a PDS.
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
2/9
Power domination
Control the world a system with a minimal number of captors [Baldwin
et al. ’91, ’93]
Some vertices in a starting set S (captors).
N [ S] = M
(propagation step) u ∈ M. If v ∈ N (u) is the only vertex outside
N [u] ∪ M: M → M ∪ {x}.
S is a power dominating set (PDS) if M = V (G) at the end.
γP (G) (power domination number of G): minimum size of a PDS.
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
2/9
Power domination
Control the world a system with a minimal number of captors [Baldwin
et al. ’91, ’93]
Some vertices in a starting set S (captors).
N [ S] = M
(propagation step) u ∈ M. If v ∈ N (u) is the only vertex outside
N [u] ∪ M: M → M ∪ {x}.
S is a power dominating set (PDS) if M = V (G) at the end.
γP (G) (power domination number of G): minimum size of a PDS.
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
2/9
Power domination
Control the world a system with a minimal number of captors [Baldwin
et al. ’91, ’93]
Some vertices in a starting set S (captors).
N [ S] = M
(propagation step) u ∈ M. If v ∈ N (u) is the only vertex outside
N [u] ∪ M: M → M ∪ {x}.
S is a power dominating set (PDS) if M = V (G) at the end.
γP (G) (power domination number of G): minimum size of a PDS.
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
2/9
Power domination
Control the world a system with a minimal number of captors [Baldwin
et al. ’91, ’93]
Some vertices in a starting set S (captors).
N [ S] = M
(propagation step) u ∈ M. If v ∈ N (u) is the only vertex outside
N [u] ∪ M: M → M ∪ {x}.
S is a power dominating set (PDS) if M = V (G) at the end.
γP (G) (power domination number of G): minimum size of a PDS.
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
2/9
Power domination
Control the world a system with a minimal number of captors [Baldwin
et al. ’91, ’93]
Some vertices in a starting set S (captors).
N [ S] = M
(propagation step) u ∈ M. If v ∈ N (u) is the only vertex outside
N [u] ∪ M: M → M ∪ {x}.
S is a power dominating set (PDS) if M = V (G) at the end.
γP (G) (power domination number of G): minimum size of a PDS.
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
2/9
Power domination
Control the world a system with a minimal number of captors [Baldwin
et al. ’91, ’93]
Some vertices in a starting set S (captors).
N [ S] = M
(propagation step) u ∈ M. If v ∈ N (u) is the only vertex outside
N [u] ∪ M: M → M ∪ {x}.
S is a power dominating set (PDS) if M = V (G) at the end.
γP (G) (power domination number of G): minimum size of a PDS.
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
2/9
Power domination
Control the world a system with a minimal number of captors [Baldwin
et al. ’91, ’93]
Some vertices in a starting set S (captors).
N [ S] = M
(propagation step) u ∈ M. If v ∈ N (u) is the only vertex outside
N [u] ∪ M: M → M ∪ {x}.
S is a power dominating set (PDS) if M = V (G) at the end.
γP (G) (power domination number of G): minimum size of a PDS.
γP (G) ≤ 3
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
2/9
Power domination
Control the world a system with a minimal number of captors [Baldwin
et al. ’91, ’93]
Some vertices in a starting set S (captors).
N [ S] = M
(propagation step) u ∈ M. If v ∈ N (u) is the only vertex outside
N [u] ∪ M: M → M ∪ {x}.
S is a power dominating set (PDS) if M = V (G) at the end.
γP (G) (power domination number of G): minimum size of a PDS.
γP (G) ≤ 3
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
2/9
Power domination
Control the world a system with a minimal number of captors [Baldwin
et al. ’91, ’93]
Some vertices in a starting set S (captors).
N [ S] = M
(propagation step) u ∈ M. If v ∈ N (u) is the only vertex outside
N [u] ∪ M: M → M ∪ {x}.
S is a power dominating set (PDS) if M = V (G) at the end.
γP (G) (power domination number of G): minimum size of a PDS.
γP (G) ≤ 3
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
2/9
Power domination
Control the world a system with a minimal number of captors [Baldwin
et al. ’91, ’93]
Some vertices in a starting set S (captors).
N [ S] = M
(propagation step) u ∈ M. If v ∈ N (u) is the only vertex outside
N [u] ∪ M: M → M ∪ {x}.
S is a power dominating set (PDS) if M = V (G) at the end.
γP (G) (power domination number of G): minimum size of a PDS.
γP (G) ≤ 3
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
2/9
Power domination
Control the world a system with a minimal number of captors [Baldwin
et al. ’91, ’93]
Some vertices in a starting set S (captors).
N [ S] = M
(propagation step) u ∈ M. If v ∈ N (u) is the only vertex outside
N [u] ∪ M: M → M ∪ {x}.
S is a power dominating set (PDS) if M = V (G) at the end.
γP (G) (power domination number of G): minimum size of a PDS.
γP (G) ≤ 3
γP (G) ≤ 2
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
2/9
Power domination
Control the world a system with a minimal number of captors [Baldwin
et al. ’91, ’93]
Some vertices in a starting set S (captors).
N [ S] = M
(propagation step) u ∈ M. If v ∈ N (u) is the only vertex outside
N [u] ∪ M: M → M ∪ {x}.
S is a power dominating set (PDS) if M = V (G) at the end.
γP (G) (power domination number of G): minimum size of a PDS.
γP ≥ 1
γP (G) ≤ 3
γP (G) ≤ 2
v
γPower-domination
P ≥1
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
2/9
Power domination
Control the world a system with a minimal number of captors [Baldwin
et al. ’91, ’93]
Some vertices in a starting set S (captors).
N [ S] = M
(propagation step) u ∈ M. If v ∈ N (u) is the only vertex outside
N [u] ∪ M: M → M ∪ {x}.
S is a power dominating set (PDS) if M = V (G) at the end.
γP (G) (power domination number of G): minimum size of a PDS.
γP ≥ 1
γP (G) ≤ 3
γP (G) ≤ 2
v
v is not sufficient
γPower-domination
P ≥1
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
2/9
Power domination
Control the world a system with a minimal number of captors [Baldwin
et al. ’91, ’93]
Some vertices in a starting set S (captors).
N [ S] = M
(propagation step) u ∈ M. If v ∈ N (u) is the only vertex outside
N [u] ∪ M: M → M ∪ {x}.
S is a power dominating set (PDS) if M = V (G) at the end.
γP (G) (power domination number of G): minimum size of a PDS.
γP ≥ 1
γP (G) ≤ 3
γP (G) ≤ 2
γP (G) = 2
v
v is not sufficient
γPower-domination
P ≥1
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
2/9
Some known results
Power-dominating set
Input: A (undirected) graph G = (V , E), an integer k ≥ 0.
Question: Is there a power-dominating set S ⊆ V with |S| ≤ k?
is NP-complete for planar graphs [Guo et al. ’05]
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
3/9
Some known results
Power-dominating set
Input: A (undirected) graph G = (V , E), an integer k ≥ 0.
Question: Is there a power-dominating set S ⊆ V with |S| ≤ k?
is NP-complete for planar graphs [Guo et al. ’05]
"Problem" with planar graphs: vertex v separating G in two connected
components G0 and G00 with δG0 (v) ≥ 2 and δG0 (v) ≥ 2.
G0
G00
v
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
3/9
Some known results
Power-dominating set
Input: A (undirected) graph G = (V , E), an integer k ≥ 0.
Question: Is there a power-dominating set S ⊆ V with |S| ≤ k?
is NP-complete for planar graphs [Guo et al. ’05]
"Problem" with planar graphs: vertex v separating G in two connected
components G0 and G00 with δG0 (v) ≥ 2 and δG0 (v) ≥ 2.
G0
G00
v
→ restrict to triangulations: no cut-vertex!
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
3/9
Power-domination in triangulations
[Matheson & Tarjan ’96]
γ(G) ≤
n
n
for (sufficiently large) triangulations of order n (conjecture: )
3
4
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
4/9
Power-domination in triangulations
[Matheson & Tarjan ’96]
n
n
for (sufficiently large) triangulations of order n (conjecture: )
3
4
n
Tight graphs with γ(G) = : each induced K4 needs a vertex in S.
4
γ(G) ≤
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
4/9
Power-domination in triangulations
[Matheson & Tarjan ’96]
n
n
for (sufficiently large) triangulations of order n (conjecture: )
3
4
n
Tight graphs with γ(G) = : each induced K4 needs a vertex in S.
4
γ(G) ≤
Power domination: propagation stops when every vertex of M "on the
boundary" has ≥ 2 neighbors in M.
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
4/9
Power-domination in triangulations
[Matheson & Tarjan ’96]
n
n
for (sufficiently large) triangulations of order n (conjecture: )
3
4
n
Tight graphs with γ(G) = : each induced K4 needs a vertex in S.
4
γ(G) ≤
n
n
n
n
n
n
Power domination: propagation stops when every vertex of M "on the
boundary" has ≥ 2 neighbors in M.
n
Tight graphs with γP (G) = : each induced octahedron needs a captor.
6
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
4/9
Power-domination in triangulations
[Matheson & Tarjan ’96]
n
n
for (sufficiently large) triangulations of order n (conjecture: )
3
4
n
Tight graphs with γ(G) = : each induced K4 needs a vertex in S.
4
γ(G) ≤
n
n
n
n
n
n
Power domination: propagation stops when every vertex of M "on the
boundary" has ≥ 2 neighbors in M.
n
Tight graphs with γP (G) = : each induced octahedron needs a captor.
6
Main Theorem
γP (G) ≤
n−2
if G is a triangulation with n ≥ 6 vertices.
4
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
4/9
Our algorithm
Monitor the
octahedrons, and
propagate.
For every vertex v in M
in decreasing degree
(in G) order: if adding v
to S adds at least 4
vertices in M (with
propagation):
add v to S.
No vertex selected:
stop.
Intuitively: monitor ≥ 6 vertices with the first captor, then ≥ 4 vertices
with each captor.
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
5/9
Our algorithm
Monitor the
octahedrons, and
propagate.
For every vertex v in M
in decreasing degree
(in G) order: if adding v
to S adds at least 4
vertices in M (with
propagation):
add v to S.
No vertex selected:
stop.
Intuitively: monitor ≥ 6 vertices with the first captor, then ≥ 4 vertices
with each captor.
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
5/9
Our algorithm
Monitor the
octahedrons, and
propagate.
For every vertex v in M
in decreasing degree
(in G) order: if adding v
to S adds at least 4
vertices in M (with
propagation):
add v to S.
No vertex selected:
stop.
Isolated octahedron:
select a vertex of the outer face in S
Intuitively: monitor ≥ 6 vertices with the first captor, then ≥ 4 vertices
with each captor.
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
5/9
Our algorithm
Monitor the
octahedrons, and
propagate.
For every vertex v in M
in decreasing degree
(in G) order: if adding v
to S adds at least 4
vertices in M (with
propagation):
add v to S.
No vertex selected:
stop.
Isolated octahedron:
select a vertex of the outer face in S
Intuitively: monitor ≥ 6 vertices with the first captor, then ≥ 4 vertices
with each captor.
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
5/9
Our algorithm
Monitor the
octahedrons, and
propagate.
For every vertex v in M
in decreasing degree
(in G) order: if adding v
to S adds at least 4
vertices in M (with
propagation):
add v to S.
No vertex selected:
stop.
Isolated octahedron:
select a vertex of the outer face in S
Intuitively: monitor ≥ 6 vertices with the first captor, then ≥ 4 vertices
with each captor.
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
5/9
Our algorithm
Monitor the
octahedrons, and
propagate.
For every vertex v in M
in decreasing degree
(in G) order: if adding v
to S adds at least 4
vertices in M (with
propagation):
add v to S.
No vertex selected:
stop.
Isolated octahedron:
select a vertex of the outer face in S
Intuitively: monitor ≥ 6 vertices with the first captor, then ≥ 4 vertices
with each captor.
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
5/9
Our algorithm
Monitor the
octahedrons, and
propagate.
For every vertex v in M
in decreasing degree
(in G) order: if adding v
to S adds at least 4
vertices in M (with
propagation):
add v to S.
No vertex selected:
stop.
2 octahedra sharing a vertex:
Select it in S
Intuitively: monitor ≥ 6 vertices with the first captor, then ≥ 4 vertices
with each captor.
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
5/9
Our algorithm
Monitor the
octahedrons, and
propagate.
For every vertex v in M
in decreasing degree
(in G) order: if adding v
to S adds at least 4
vertices in M (with
propagation):
add v to S.
No vertex selected:
stop.
2 octahedra sharing a vertex:
Select it in S
Intuitively: monitor ≥ 6 vertices with the first captor, then ≥ 4 vertices
with each captor.
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
5/9
Our algorithm
Monitor the
octahedrons, and
propagate.
For every vertex v in M
in decreasing degree
(in G) order: if adding v
to S adds at least 4
vertices in M (with
propagation):
add v to S.
No vertex selected:
stop.
2 octahedra sharing a vertex:
Select it in S
Intuitively: monitor ≥ 6 vertices with the first captor, then ≥ 4 vertices
with each captor.
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
5/9
Our algorithm
Monitor the
octahedrons, and
propagate.
For every vertex v in M
in decreasing degree
(in G) order: if adding v
to S adds at least 4
vertices in M (with
propagation):
add v to S.
No vertex selected:
stop.
2 octahedra sharing a vertex:
Select it in S
Intuitively: monitor ≥ 6 vertices with the first captor, then ≥ 4 vertices
with each captor.
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
5/9
Our algorithm
Monitor the
octahedrons, and
propagate.
For every vertex v in M
in decreasing degree
(in G) order: if adding v
to S adds at least 4
vertices in M (with
propagation):
add v to S.
No vertex selected:
stop.
≥ 6 new
vertices in M
2 octahedra sharing a vertex:
Select it in S
Intuitively: monitor ≥ 6 vertices with the first captor, then ≥ 4 vertices
with each captor.
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
5/9
Our algorithm
Monitor the
octahedrons, and
propagate.
For every vertex v in M
in decreasing degree
(in G) order: if adding v
to S adds at least 4
vertices in M (with
propagation):
add v to S.
No vertex selected:
stop.
Intuitively: monitor ≥ 6 vertices with the first captor, then ≥ 4 vertices
with each captor.
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
5/9
Our algorithm
Monitor the
octahedrons, and
propagate.
For every vertex v in M
in decreasing degree
(in G) order: if adding v
to S adds at least 4
vertices in M (with
propagation):
add v to S.
No vertex selected:
stop.
Intuitively: monitor ≥ 6 vertices with the first captor, then ≥ 4 vertices
with each captor.
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
5/9
Our algorithm
Monitor the
octahedrons, and
propagate.
For every vertex v in M
in decreasing degree
(in G) order: if adding v
to S adds at least 4
vertices in M (with
propagation):
add v to S.
No vertex selected:
stop.
Intuitively: monitor ≥ 6 vertices with the first captor, then ≥ 4 vertices
with each captor.
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
5/9
Our algorithm
Monitor the
octahedrons, and
propagate.
For every vertex v in M
in decreasing degree
(in G) order: if adding v
to S adds at least 4
vertices in M (with
propagation):
add v to S.
No vertex selected:
stop.
Intuitively: monitor ≥ 6 vertices with the first captor, then ≥ 4 vertices
with each captor.
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
5/9
Our algorithm
Monitor the
octahedrons, and
propagate.
For every vertex v in M
in decreasing degree
(in G) order: if adding v
to S adds at least 4
vertices in M (with
propagation):
add v to S.
No vertex selected:
stop.
Intuitively: monitor ≥ 6 vertices with the first captor, then ≥ 4 vertices
with each captor.
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
5/9
Our algorithm
Monitor the
octahedrons, and
propagate.
For every vertex v in M
in decreasing degree
(in G) order: if adding v
to S adds at least 4
vertices in M (with
propagation):
add v to S.
No vertex selected:
stop.
Intuitively: monitor ≥ 6 vertices with the first captor, then ≥ 4 vertices
with each captor.
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
5/9
Our algorithm
Monitor the
octahedrons, and
propagate.
For every vertex v in M
in decreasing degree
(in G) order: if adding v
to S adds at least 4
vertices in M (with
propagation):
add v to S.
No vertex selected:
stop.
Intuitively: monitor ≥ 6 vertices with the first captor, then ≥ 4 vertices
with each captor.
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
5/9
Our algorithm
Monitor the
octahedrons, and
propagate.
For every vertex v in M
in decreasing degree
(in G) order: if adding v
to S adds at least 4
vertices in M (with
propagation):
add v to S.
No vertex selected:
stop.
Intuitively: monitor ≥ 6 vertices with the first captor, then ≥ 4 vertices
with each captor.
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
5/9
Our algorithm
Monitor the
octahedrons, and
propagate.
For every vertex v in M
in decreasing degree
(in G) order: if adding v
to S adds at least 4
vertices in M (with
propagation):
add v to S.
No vertex selected:
stop.
Intuitively: monitor ≥ 6 vertices with the first captor, then ≥ 4 vertices
with each captor.
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
5/9
Our algorithm
Monitor the
octahedrons, and
propagate.
For every vertex v in M
in decreasing degree
(in G) order: if adding v
to S adds at least 4
vertices in M (with
propagation):
add v to S.
No vertex selected:
stop.
Intuitively: monitor ≥ 6 vertices with the first captor, then ≥ 4 vertices
with each captor.
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
5/9
Our algorithm
Monitor the
octahedrons, and
propagate.
For every vertex v in M
in decreasing degree
(in G) order: if adding v
to S adds at least 4
vertices in M (with
propagation):
add v to S.
No vertex selected:
stop.
Intuitively: monitor ≥ 6 vertices with the first captor, then ≥ 4 vertices
with each captor.
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
5/9
Our algorithm
Monitor the
octahedrons, and
propagate.
For every vertex v in M
in decreasing degree
(in G) order: if adding v
to S adds at least 4
vertices in M (with
propagation):
add v to S.
No vertex selected:
stop.
Intuitively: monitor ≥ 6 vertices with the first captor, then ≥ 4 vertices
with each captor.
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
5/9
The 4 cases
The graph G[M] satisfies the following properties:
(a) G[M] has maximum degree at most 2.
(b) Each connected component of G[M] has at most three vertices.
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
6/9
The 4 cases
The graph G[M] satisfies the following properties:
(a) G[M] has maximum degree at most 2.
(Otherwise, if u has degree ≥ 3, take u in S)
(b) Each connected component of G[M] has at most three vertices.
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
6/9
The 4 cases
The graph G[M] satisfies the following properties:
(a) G[M] has maximum degree at most 2.
(Otherwise, if u has degree ≥ 3, take u in S)
(b) Each connected component of G[M] has at most three vertices.
(Each cc is a cycle or a path. If ≥ 4 vertices: any vertex propagates
to 3 others)
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
6/9
The 4 cases
The graph G[M] satisfies the following properties:
(a) G[M] has maximum degree at most 2.
(Otherwise, if u has degree ≥ 3, take u in S)
(b) Each connected component of G[M] has at most three vertices.
(Each cc is a cycle or a path. If ≥ 4 vertices: any vertex propagates
to 3 others)
→ A connected component of G[M] is isomorphic to K3 , P3 , P2 or K1 .
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
6/9
The 4 cases
The graph G[M] satisfies the following properties:
(a) G[M] has maximum degree at most 2.
(Otherwise, if u has degree ≥ 3, take u in S)
(b) Each connected component of G[M] has at most three vertices.
(Each cc is a cycle or a path. If ≥ 4 vertices: any vertex propagates
to 3 others)
→ A connected component of G[M] is isomorphic to K3 , P3 , P2 or K1 .
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
6/9
The 4 cases
The graph G[M] satisfies the following properties:
(a) G[M] has maximum degree at most 2.
(Otherwise, if u has degree ≥ 3, take u in S)
(b) Each connected component of G[M] has at most three vertices.
(Each cc is a cycle or a path. If ≥ 4 vertices: any vertex propagates
to 3 others)
→ A connected component of G[M] is isomorphic to K3 , P3 , P2 or K1 .
lead to unique configurations
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
6/9
Connected components of G[M]
Global technique used for all cases: try to build G around the
hypothetical connected component.
(Some) Tools used in this (long) proof:
planarity (contradiction with Euler’s formula)
contradiction with the conditions to choose a vertex in S :
maximal degree or contribution of each vertex
induction reasoning
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
7/9
Induction example: P3
If a connected component of G[M] is isomorphic to P3 , then G is
isomorphic to:
y
G1
w
x1
z
x3
x2
w0
z0
G2
y0
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
8/9
Induction example: P3
If a connected component of G[M] is isomorphic to P3 , then G is
isomorphic to:
G1 and G2 have ≥ 6 vertices (oth.
contradiction with the degree
condition)
y
G1
w
x1
z
x3
x2
w0
z0
G2
y0
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
8/9
Induction example: P3
If a connected component of G[M] is isomorphic to P3 , then G is
isomorphic to:
G1 and G2 have ≥ 6 vertices (oth.
contradiction with the degree
condition)
y
G1
w
x1
z
x3
x2
w0
Induction on the size:
z0
G2
y0
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
8/9
Induction example: P3
If a connected component of G[M] is isomorphic to P3 , then G is
isomorphic to:
G1 and G2 have ≥ 6 vertices (oth.
contradiction with the degree
condition)
y
G1
w
x1
z
x3
x2
w0
Induction on the size:
n1 − 2
γP (G1 ) ≤
4
z0
G2
y0
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
8/9
Induction example: P3
If a connected component of G[M] is isomorphic to P3 , then G is
isomorphic to:
G1 and G2 have ≥ 6 vertices (oth.
contradiction with the degree
condition)
y
G1
w
x1
z
x3
x2
w0
Induction on the size:
n1 − 2
γP (G1 ) ≤
4
n2 − 2
γP (G2 ) ≤
4
z0
G2
y0
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
8/9
Induction example: P3
If a connected component of G[M] is isomorphic to P3 , then G is
isomorphic to:
G1 and G2 have ≥ 6 vertices (oth.
contradiction with the degree
condition)
y
G1
w
z
x1
x3
x2
w
0
z0
G2
y0
Induction on the size:
n1 − 2
γP (G1 ) ≤
4
n2 − 2
γP (G2 ) ≤
4
Adding x2 to S:
n1 + n2 − 4
n1 + n2
+1 =
γP (G) ≤
4
4
n1 + n2 n − 2
and
<
4
4
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
8/9
Induction example: P3
If a connected component of G[M] is isomorphic to P3 , then G is
isomorphic to:
G1 and G2 have ≥ 6 vertices (oth.
contradiction with the degree
condition)
y
G1
w
z
x1
x3
x2
w
0
z0
G2
y0
Induction on the size:
n1 − 2
γP (G1 ) ≤
4
n2 − 2
γP (G2 ) ≤
4
Adding x2 to S:
n1 + n2 − 4
n1 + n2
+1 =
γP (G) ≤
4
4
n1 + n2 n − 2
and
<
4
4
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
8/9
Open questions
n−2
n
? (the lower bound is ...)
4
6
Is the decision problem NP-Complete for triangulations?
Can we do better than
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
9/9
Open questions
n−2
n
? (the lower bound is ...)
4
6
Is the decision problem NP-Complete for triangulations?
Can we do better than
Thank you!
Claire Pennarun (LaBRI, Université de Bordeaux Universidad
Power-domination
de Cadiz)
in triangulations
JGA, Orléans, 6 novembre 2015
9/9