domination and roman domination in some product …introduction domination in graphs g = (v;e), a...
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DOMINATION AND ROMAN DOMINATION INSOME PRODUCT GRAPHS
ISMAEL GONZALEZ YERO
Departamento de Matematicas,Universidad de Cadiz - Escuela Politecnica Superior de Algeciras,
Av. Ramon Puyol s/n, 11202 Algeciras, EspanaE-mail: [email protected]
Joint work with D. Kuziak and J. A. Rodrıguez-Velazquez
igyero (EPS-UCA) GRAPHS 1 / 24
Index
Index
1 Introduction
2 Cartesian product graphs
3 Strong product graphs
4 Rooted product graphs
igyero (EPS-UCA) GRAPHS 2 / 24
Index
Index
1 Introduction
2 Cartesian product graphs
3 Strong product graphs
4 Rooted product graphs
igyero (EPS-UCA) GRAPHS 2 / 24
Index
Index
1 Introduction
2 Cartesian product graphs
3 Strong product graphs
4 Rooted product graphs
igyero (EPS-UCA) GRAPHS 2 / 24
Index
Index
1 Introduction
2 Cartesian product graphs
3 Strong product graphs
4 Rooted product graphs
igyero (EPS-UCA) GRAPHS 2 / 24
Introduction
Index
1 Introduction
2 Cartesian product graphs
3 Strong product graphs
4 Rooted product graphs
igyero (EPS-UCA) GRAPHS 3 / 24
Introduction
Domination in graphs
G = (V ,E ), a simple graph. S ⊂ V , set of vertices of G .
S is a dominating set if N(S) = V , i.e., every vertex v ∈ S isadjacent to a vertex of S .
γ(G ), domination number of G : minimum cardinality of anydominating set in G .
Domination related parameters
Domination plus conditions on vertices of the dominating set or itscomplement: Total domination, connected domination, independentdomination, etc.
Conditions over the style of domination: k-domination, distancedomination, etc.
Dominating functions: Roman domination, signed domination, minusdomination, etc.
igyero (EPS-UCA) GRAPHS 4 / 24
Introduction
Domination in graphs
G = (V ,E ), a simple graph. S ⊂ V , set of vertices of G .
S is a dominating set if N(S) = V , i.e., every vertex v ∈ S isadjacent to a vertex of S .
γ(G ), domination number of G : minimum cardinality of anydominating set in G .
Domination related parameters
Domination plus conditions on vertices of the dominating set or itscomplement: Total domination, connected domination, independentdomination, etc.
Conditions over the style of domination: k-domination, distancedomination, etc.
Dominating functions: Roman domination, signed domination, minusdomination, etc.
igyero (EPS-UCA) GRAPHS 4 / 24
Introduction
Domination in graphs
G = (V ,E ), a simple graph. S ⊂ V , set of vertices of G .
S is a dominating set if N(S) = V , i.e., every vertex v ∈ S isadjacent to a vertex of S .
γ(G ), domination number of G : minimum cardinality of anydominating set in G .
Domination related parameters
Domination plus conditions on vertices of the dominating set or itscomplement: Total domination, connected domination, independentdomination, etc.
Conditions over the style of domination: k-domination, distancedomination, etc.
Dominating functions: Roman domination, signed domination, minusdomination, etc.
igyero (EPS-UCA) GRAPHS 4 / 24
Introduction
Domination in graphs
G = (V ,E ), a simple graph. S ⊂ V , set of vertices of G .
S is a dominating set if N(S) = V , i.e., every vertex v ∈ S isadjacent to a vertex of S .
γ(G ), domination number of G : minimum cardinality of anydominating set in G .
Domination related parameters
Domination plus conditions on vertices of the dominating set or itscomplement: Total domination, connected domination, independentdomination, etc.
Conditions over the style of domination: k-domination, distancedomination, etc.
Dominating functions: Roman domination, signed domination, minusdomination, etc.
igyero (EPS-UCA) GRAPHS 4 / 24
Introduction
Domination in graphs
G = (V ,E ), a simple graph. S ⊂ V , set of vertices of G .
S is a dominating set if N(S) = V , i.e., every vertex v ∈ S isadjacent to a vertex of S .
γ(G ), domination number of G : minimum cardinality of anydominating set in G .
Domination related parameters
Domination plus conditions on vertices of the dominating set or itscomplement: Total domination, connected domination, independentdomination, etc.
Conditions over the style of domination: k-domination, distancedomination, etc.
Dominating functions: Roman domination, signed domination, minusdomination, etc.
igyero (EPS-UCA) GRAPHS 4 / 24
Introduction
Domination in graphs
G = (V ,E ), a simple graph. S ⊂ V , set of vertices of G .
S is a dominating set if N(S) = V , i.e., every vertex v ∈ S isadjacent to a vertex of S .
γ(G ), domination number of G : minimum cardinality of anydominating set in G .
Domination related parameters
Domination plus conditions on vertices of the dominating set or itscomplement: Total domination, connected domination, independentdomination, etc.
Conditions over the style of domination: k-domination, distancedomination, etc.
Dominating functions: Roman domination, signed domination, minusdomination, etc.
igyero (EPS-UCA) GRAPHS 4 / 24
Introduction
Domination in graphs
G = (V ,E ), a simple graph. S ⊂ V , set of vertices of G .
S is a dominating set if N(S) = V , i.e., every vertex v ∈ S isadjacent to a vertex of S .
γ(G ), domination number of G : minimum cardinality of anydominating set in G .
Domination related parameters
Domination plus conditions on vertices of the dominating set or itscomplement: Total domination, connected domination, independentdomination, etc.
Conditions over the style of domination: k-domination, distancedomination, etc.
Dominating functions: Roman domination, signed domination, minusdomination, etc.
igyero (EPS-UCA) GRAPHS 4 / 24
Introduction
Domination in graphs
G = (V ,E ), a simple graph. S ⊂ V , set of vertices of G .
S is a dominating set if N(S) = V , i.e., every vertex v ∈ S isadjacent to a vertex of S .
γ(G ), domination number of G : minimum cardinality of anydominating set in G .
Domination related parameters
Domination plus conditions on vertices of the dominating set or itscomplement: Total domination, connected domination, independentdomination, etc.
Conditions over the style of domination: k-domination, distancedomination, etc.
Dominating functions: Roman domination, signed domination, minusdomination, etc.
igyero (EPS-UCA) GRAPHS 4 / 24
Introduction
Roman domination
A map f : V → {0, 1, 2}, Roman dominating function for G , if forevery v ∈ V with f (v) = 0 there exists u ∈ N(v) such that f (u) = 2.
The weight of f if f (V ) =∑
v∈V f (v).
γR(G ), Roman domination number of G : minimum weight of anyRoman dominating function for G .
Every Roman dominating function induces three sets B0,B1,B2 suchthat Bi = {v ∈ V : f (v) = i}, i ∈ {0, 1, 2}.
γ(G ) ≤ γR(G ) ≤ 2γ(G )
γ(G ) = γR(G ) if and only if G = Kn.
G is called a Roman graph if γR(G ) = 2γ(G ).
There is an open problem related to characterizing all Roman graphs(Roman trees are characterized).
igyero (EPS-UCA) GRAPHS 5 / 24
Introduction
Roman domination
A map f : V → {0, 1, 2}, Roman dominating function for G , if forevery v ∈ V with f (v) = 0 there exists u ∈ N(v) such that f (u) = 2.
The weight of f if f (V ) =∑
v∈V f (v).
γR(G ), Roman domination number of G : minimum weight of anyRoman dominating function for G .
Every Roman dominating function induces three sets B0,B1,B2 suchthat Bi = {v ∈ V : f (v) = i}, i ∈ {0, 1, 2}.
γ(G ) ≤ γR(G ) ≤ 2γ(G )
γ(G ) = γR(G ) if and only if G = Kn.
G is called a Roman graph if γR(G ) = 2γ(G ).
There is an open problem related to characterizing all Roman graphs(Roman trees are characterized).
igyero (EPS-UCA) GRAPHS 5 / 24
Introduction
Roman domination
A map f : V → {0, 1, 2}, Roman dominating function for G , if forevery v ∈ V with f (v) = 0 there exists u ∈ N(v) such that f (u) = 2.
The weight of f if f (V ) =∑
v∈V f (v).
γR(G ), Roman domination number of G : minimum weight of anyRoman dominating function for G .
Every Roman dominating function induces three sets B0,B1,B2 suchthat Bi = {v ∈ V : f (v) = i}, i ∈ {0, 1, 2}.
γ(G ) ≤ γR(G ) ≤ 2γ(G )
γ(G ) = γR(G ) if and only if G = Kn.
G is called a Roman graph if γR(G ) = 2γ(G ).
There is an open problem related to characterizing all Roman graphs(Roman trees are characterized).
igyero (EPS-UCA) GRAPHS 5 / 24
Introduction
Roman domination
A map f : V → {0, 1, 2}, Roman dominating function for G , if forevery v ∈ V with f (v) = 0 there exists u ∈ N(v) such that f (u) = 2.
The weight of f if f (V ) =∑
v∈V f (v).
γR(G ), Roman domination number of G : minimum weight of anyRoman dominating function for G .
Every Roman dominating function induces three sets B0,B1,B2 suchthat Bi = {v ∈ V : f (v) = i}, i ∈ {0, 1, 2}.
γ(G ) ≤ γR(G ) ≤ 2γ(G )
γ(G ) = γR(G ) if and only if G = Kn.
G is called a Roman graph if γR(G ) = 2γ(G ).
There is an open problem related to characterizing all Roman graphs(Roman trees are characterized).
igyero (EPS-UCA) GRAPHS 5 / 24
Introduction
Roman domination
A map f : V → {0, 1, 2}, Roman dominating function for G , if forevery v ∈ V with f (v) = 0 there exists u ∈ N(v) such that f (u) = 2.
The weight of f if f (V ) =∑
v∈V f (v).
γR(G ), Roman domination number of G : minimum weight of anyRoman dominating function for G .
Every Roman dominating function induces three sets B0,B1,B2 suchthat Bi = {v ∈ V : f (v) = i}, i ∈ {0, 1, 2}.
γ(G ) ≤ γR(G ) ≤ 2γ(G )
γ(G ) = γR(G ) if and only if G = Kn.
G is called a Roman graph if γR(G ) = 2γ(G ).
There is an open problem related to characterizing all Roman graphs(Roman trees are characterized).
igyero (EPS-UCA) GRAPHS 5 / 24
Introduction
Roman domination
A map f : V → {0, 1, 2}, Roman dominating function for G , if forevery v ∈ V with f (v) = 0 there exists u ∈ N(v) such that f (u) = 2.
The weight of f if f (V ) =∑
v∈V f (v).
γR(G ), Roman domination number of G : minimum weight of anyRoman dominating function for G .
Every Roman dominating function induces three sets B0,B1,B2 suchthat Bi = {v ∈ V : f (v) = i}, i ∈ {0, 1, 2}.
γ(G ) ≤ γR(G ) ≤ 2γ(G )
γ(G ) = γR(G ) if and only if G = Kn.
G is called a Roman graph if γR(G ) = 2γ(G ).
There is an open problem related to characterizing all Roman graphs(Roman trees are characterized).
igyero (EPS-UCA) GRAPHS 5 / 24
Introduction
Roman domination
A map f : V → {0, 1, 2}, Roman dominating function for G , if forevery v ∈ V with f (v) = 0 there exists u ∈ N(v) such that f (u) = 2.
The weight of f if f (V ) =∑
v∈V f (v).
γR(G ), Roman domination number of G : minimum weight of anyRoman dominating function for G .
Every Roman dominating function induces three sets B0,B1,B2 suchthat Bi = {v ∈ V : f (v) = i}, i ∈ {0, 1, 2}.
γ(G ) ≤ γR(G ) ≤ 2γ(G )
γ(G ) = γR(G ) if and only if G = Kn.
G is called a Roman graph if γR(G ) = 2γ(G ).
There is an open problem related to characterizing all Roman graphs(Roman trees are characterized).
igyero (EPS-UCA) GRAPHS 5 / 24
Introduction
Roman domination
A map f : V → {0, 1, 2}, Roman dominating function for G , if forevery v ∈ V with f (v) = 0 there exists u ∈ N(v) such that f (u) = 2.
The weight of f if f (V ) =∑
v∈V f (v).
γR(G ), Roman domination number of G : minimum weight of anyRoman dominating function for G .
Every Roman dominating function induces three sets B0,B1,B2 suchthat Bi = {v ∈ V : f (v) = i}, i ∈ {0, 1, 2}.
γ(G ) ≤ γR(G ) ≤ 2γ(G )
γ(G ) = γR(G ) if and only if G = Kn.
G is called a Roman graph if γR(G ) = 2γ(G ).
There is an open problem related to characterizing all Roman graphs(Roman trees are characterized).
igyero (EPS-UCA) GRAPHS 5 / 24
Introduction
Roman domination
A map f : V → {0, 1, 2}, Roman dominating function for G , if forevery v ∈ V with f (v) = 0 there exists u ∈ N(v) such that f (u) = 2.
The weight of f if f (V ) =∑
v∈V f (v).
γR(G ), Roman domination number of G : minimum weight of anyRoman dominating function for G .
Every Roman dominating function induces three sets B0,B1,B2 suchthat Bi = {v ∈ V : f (v) = i}, i ∈ {0, 1, 2}.
γ(G ) ≤ γR(G ) ≤ 2γ(G )
γ(G ) = γR(G ) if and only if G = Kn.
G is called a Roman graph if γR(G ) = 2γ(G ).
There is an open problem related to characterizing all Roman graphs(Roman trees are characterized).
igyero (EPS-UCA) GRAPHS 5 / 24
Introduction
Product graphs
Cartesian product graphs, G�H
Direct product graphs, G × H
Strong product graphs, G � H Rooted product graphs, G ◦ H
igyero (EPS-UCA) GRAPHS 6 / 24
Introduction
Product graphs
Cartesian product graphs, G�H Direct product graphs, G × H
Strong product graphs, G � H Rooted product graphs, G ◦ H
igyero (EPS-UCA) GRAPHS 6 / 24
Introduction
Product graphs
Cartesian product graphs, G�H Direct product graphs, G × H
Strong product graphs, G � H
Rooted product graphs, G ◦ H
igyero (EPS-UCA) GRAPHS 6 / 24
Introduction
Product graphs
Cartesian product graphs, G�H Direct product graphs, G × H
Strong product graphs, G � H Rooted product graphs, G ◦ H
igyero (EPS-UCA) GRAPHS 6 / 24
Introduction
Domination versus product graphs
Vizing’s conjecture
One of the most important problems about domination in graphs:γ(G�H) ≥ γ(G )γ(H).
Several Vizing-like results for other domination (also not dominationrelated) parameters.
Γ(G�H) ≥ Γ(G )Γ(H), γ(G × H) ≤ 3γ(G )γ(H),γ(G � H) ≤ γ(G )γ(H), etc.
The best approximation to Vizing’s conjecture:2γ(G�H) ≥ γ(G )γ(H) (Clark and Suen).
Roman domination
γR(G�H) ≥ γ(G )γ(H).
There were no more results in this topic.
So, we did it.
igyero (EPS-UCA) GRAPHS 7 / 24
Introduction
Domination versus product graphs
Vizing’s conjecture
One of the most important problems about domination in graphs:γ(G�H) ≥ γ(G )γ(H).
Several Vizing-like results for other domination (also not dominationrelated) parameters.
Γ(G�H) ≥ Γ(G )Γ(H), γ(G × H) ≤ 3γ(G )γ(H),γ(G � H) ≤ γ(G )γ(H), etc.
The best approximation to Vizing’s conjecture:2γ(G�H) ≥ γ(G )γ(H) (Clark and Suen).
Roman domination
γR(G�H) ≥ γ(G )γ(H).
There were no more results in this topic.
So, we did it.
igyero (EPS-UCA) GRAPHS 7 / 24
Introduction
Domination versus product graphs
Vizing’s conjecture
One of the most important problems about domination in graphs:γ(G�H) ≥ γ(G )γ(H).
Several Vizing-like results for other domination (also not dominationrelated) parameters.
Γ(G�H) ≥ Γ(G )Γ(H), γ(G × H) ≤ 3γ(G )γ(H),γ(G � H) ≤ γ(G )γ(H), etc.
The best approximation to Vizing’s conjecture:2γ(G�H) ≥ γ(G )γ(H) (Clark and Suen).
Roman domination
γR(G�H) ≥ γ(G )γ(H).
There were no more results in this topic.
So, we did it.
igyero (EPS-UCA) GRAPHS 7 / 24
Introduction
Domination versus product graphs
Vizing’s conjecture
One of the most important problems about domination in graphs:γ(G�H) ≥ γ(G )γ(H).
Several Vizing-like results for other domination (also not dominationrelated) parameters.
Γ(G�H) ≥ Γ(G )Γ(H), γ(G × H) ≤ 3γ(G )γ(H),γ(G � H) ≤ γ(G )γ(H), etc.
The best approximation to Vizing’s conjecture:2γ(G�H) ≥ γ(G )γ(H) (Clark and Suen).
Roman domination
γR(G�H) ≥ γ(G )γ(H).
There were no more results in this topic.
So, we did it.
igyero (EPS-UCA) GRAPHS 7 / 24
Introduction
Domination versus product graphs
Vizing’s conjecture
One of the most important problems about domination in graphs:γ(G�H) ≥ γ(G )γ(H).
Several Vizing-like results for other domination (also not dominationrelated) parameters.
Γ(G�H) ≥ Γ(G )Γ(H), γ(G × H) ≤ 3γ(G )γ(H),γ(G � H) ≤ γ(G )γ(H), etc.
The best approximation to Vizing’s conjecture:2γ(G�H) ≥ γ(G )γ(H) (Clark and Suen).
Roman domination
γR(G�H) ≥ γ(G )γ(H).
There were no more results in this topic.
So, we did it.
igyero (EPS-UCA) GRAPHS 7 / 24
Introduction
Domination versus product graphs
Vizing’s conjecture
One of the most important problems about domination in graphs:γ(G�H) ≥ γ(G )γ(H).
Several Vizing-like results for other domination (also not dominationrelated) parameters.
Γ(G�H) ≥ Γ(G )Γ(H), γ(G × H) ≤ 3γ(G )γ(H),γ(G � H) ≤ γ(G )γ(H), etc.
The best approximation to Vizing’s conjecture:2γ(G�H) ≥ γ(G )γ(H) (Clark and Suen).
Roman domination
γR(G�H) ≥ γ(G )γ(H).
There were no more results in this topic.
So, we did it.
igyero (EPS-UCA) GRAPHS 7 / 24
Introduction
Domination versus product graphs
Vizing’s conjecture
One of the most important problems about domination in graphs:γ(G�H) ≥ γ(G )γ(H).
Several Vizing-like results for other domination (also not dominationrelated) parameters.
Γ(G�H) ≥ Γ(G )Γ(H), γ(G × H) ≤ 3γ(G )γ(H),γ(G � H) ≤ γ(G )γ(H), etc.
The best approximation to Vizing’s conjecture:2γ(G�H) ≥ γ(G )γ(H) (Clark and Suen).
Roman domination
γR(G�H) ≥ γ(G )γ(H).
There were no more results in this topic.
So, we did it.
igyero (EPS-UCA) GRAPHS 7 / 24
Introduction
Domination versus product graphs
Vizing’s conjecture
One of the most important problems about domination in graphs:γ(G�H) ≥ γ(G )γ(H).
Several Vizing-like results for other domination (also not dominationrelated) parameters.
Γ(G�H) ≥ Γ(G )Γ(H), γ(G × H) ≤ 3γ(G )γ(H),γ(G � H) ≤ γ(G )γ(H), etc.
The best approximation to Vizing’s conjecture:2γ(G�H) ≥ γ(G )γ(H) (Clark and Suen).
Roman domination
γR(G�H) ≥ γ(G )γ(H).
There were no more results in this topic.
So, we did it.
igyero (EPS-UCA) GRAPHS 7 / 24
Cartesian product graphs
Index
1 Introduction
2 Cartesian product graphs
3 Strong product graphs
4 Rooted product graphs
igyero (EPS-UCA) GRAPHS 8 / 24
Cartesian product graphs
A general bound
V1 and V2, the vertex sets ofG and H, respectively.
igyero (EPS-UCA) GRAPHS 9 / 24
Cartesian product graphs
A general bound
V1 and V2, the vertex sets ofG and H, respectively.
S = {u1, ..., ut},a dominatingset for G , t = γ(G ).
igyero (EPS-UCA) GRAPHS 10 / 24
Cartesian product graphs
A general bound
V1 and V2, the vertex sets ofG and H, respectively.
S = {u1, ..., ut},a dominatingset for G , t = γ(G ).
Π = {A1,A2, ...,Aγ(G)}, avertex partition of G such thatui ∈ Ai and Ai ⊆ N[ui ]
igyero (EPS-UCA) GRAPHS 11 / 24
Cartesian product graphs
A general bound
{Π1,Π2, ...,Πγ(G)}, a vertexpartition of G�H, such thatΠi = Ai × V2 for everyi ∈ {1, ..., γ(G )}
f = (B0,B1,B2), aγR(G�H)-function
igyero (EPS-UCA) GRAPHS 12 / 24
Cartesian product graphs
A general bound
{Π1,Π2, ...,Πγ(G)}, a vertexpartition of G�H, such thatΠi = Ai × V2 for everyi ∈ {1, ..., γ(G )}
f = (B0,B1,B2), aγR(G�H)-function
igyero (EPS-UCA) GRAPHS 12 / 24
Cartesian product graphs
A general bound
{Π1,Π2, ...,Πγ(G)}, a vertexpartition of G�H, such thatΠi = Ai × V2 for everyi ∈ {1, ..., γ(G )}
f = (B0,B1,B2), aγR(G�H)-function
For every i ∈ {1, ..., γ(G )},fi : V2 → {0, 1, 2}, a functionon H such that fi (v) =max{f (u, v) : u ∈ Ai}.
igyero (EPS-UCA) GRAPHS 13 / 24
Cartesian product graphs
A general bound
fi = (X(i)0 ,X
(i)1 ,X
(i)2 ), not a
Roman dominating function for
H, there is a vertex v ∈ X(i)0 ,
N(v) ∩ X(i)2 = ∅.
igyero (EPS-UCA) GRAPHS 14 / 24
Cartesian product graphs
A general bound
fi = (X(i)0 ,X
(i)1 ,X
(i)2 ), not a
Roman dominating function for
H, there is a vertex v ∈ X(i)0 ,
N(v) ∩ X(i)2 = ∅.
For every u ∈ Ai , (u, v) isadjacent to some vertex not inΠi
igyero (EPS-UCA) GRAPHS 15 / 24
Cartesian product graphs
A general bound
fi = (X(i)0 ,X
(i)1 ,X
(i)2 ), not a
Roman dominating function for
H, there is a vertex v ∈ X(i)0 ,
N(v) ∩ X(i)2 = ∅.
For every u ∈ Ai , (u, v) isadjacent to some vertex not inΠi
igyero (EPS-UCA) GRAPHS 15 / 24
Cartesian product graphs
A general bound
For every i ∈ {1, ..., γ(G )}, we count the number of vertices of Hsatisfying that they are not adjacent to any vertex with label two (2).
For every vertex v of H we count the G -cells satisfying that all theirvertices are not adjacent to any vertex with label two (2) in the same“column”.
By doing a double sum we get that
γR(G�H) ≥ 2
3γ(G )γR(H)
igyero (EPS-UCA) GRAPHS 16 / 24
Cartesian product graphs
A general bound
For every i ∈ {1, ..., γ(G )}, we count the number of vertices of Hsatisfying that they are not adjacent to any vertex with label two (2).
For every vertex v of H we count the G -cells satisfying that all theirvertices are not adjacent to any vertex with label two (2) in the same“column”.
By doing a double sum we get that
γR(G�H) ≥ 2
3γ(G )γR(H)
igyero (EPS-UCA) GRAPHS 16 / 24
Cartesian product graphs
A general bound
For every i ∈ {1, ..., γ(G )}, we count the number of vertices of Hsatisfying that they are not adjacent to any vertex with label two (2).
For every vertex v of H we count the G -cells satisfying that all theirvertices are not adjacent to any vertex with label two (2) in the same“column”.
By doing a double sum we get that
γR(G�H) ≥ 2
3γ(G )γR(H)
igyero (EPS-UCA) GRAPHS 16 / 24
Cartesian product graphs
The general bound
For any graphs G and H,
γR(G�H) ≥ 2
3γ(G )γR(H).
If γR(H) > 3γ(H)2 , then
γ(G�H) ≥ γ(G )γ(H)
2+γ(G )
3.
For any graph G and any Roman graph H,
γR(G�H) ≥ 4
3γ(G )γ(H).
γ(G�H) ≥ 2
3γ(G )γ(H).
igyero (EPS-UCA) GRAPHS 17 / 24
Cartesian product graphs
The general bound
For any graphs G and H,
γR(G�H) ≥ 2
3γ(G )γR(H).
If γR(H) > 3γ(H)2 , then
γ(G�H) ≥ γ(G )γ(H)
2+γ(G )
3.
For any graph G and any Roman graph H,
γR(G�H) ≥ 4
3γ(G )γ(H).
γ(G�H) ≥ 2
3γ(G )γ(H).
igyero (EPS-UCA) GRAPHS 17 / 24
Cartesian product graphs
The general bound
For any graphs G and H,
γR(G�H) ≥ 2
3γ(G )γR(H).
If γR(H) > 3γ(H)2 , then
γ(G�H) ≥ γ(G )γ(H)
2+γ(G )
3.
For any graph G and any Roman graph H,
γR(G�H) ≥ 4
3γ(G )γ(H).
γ(G�H) ≥ 2
3γ(G )γ(H).
igyero (EPS-UCA) GRAPHS 17 / 24
Strong product graphs
Index
1 Introduction
2 Cartesian product graphs
3 Strong product graphs
4 Rooted product graphs
igyero (EPS-UCA) GRAPHS 18 / 24
Strong product graphs
Roman domination
f1 = (A0,A1,A2), γR(G )-function. f2 = (B0,B1,B2), γR(H)-function.Then,
γR(G � H) ≤ γR(G )γR(H)− 2|A2||B2|.
Idea of the proof
f on G � H defined as
f (u, v) =
2, (u, v) ∈ (A1 × B2) ∪ (A2 × B1) ∪ (A2 × B2),1, (u, v) ∈ A1 × B1,0, otherwise.
(A0 × B0) ∪ (A0 × B2) ∪ (A2 × B0) is dominated by A2 × B2,
A1 × B0 is dominated by A1 × B2 and
A0 × B1 is dominated by A2 × B1.
f is a Roman dominating function on G � H.
igyero (EPS-UCA) GRAPHS 19 / 24
Strong product graphs
Roman domination
f1 = (A0,A1,A2), γR(G )-function. f2 = (B0,B1,B2), γR(H)-function.Then,
γR(G � H) ≤ γR(G )γR(H)− 2|A2||B2|.
Idea of the proof
f on G � H defined as
f (u, v) =
2, (u, v) ∈ (A1 × B2) ∪ (A2 × B1) ∪ (A2 × B2),1, (u, v) ∈ A1 × B1,0, otherwise.
(A0 × B0) ∪ (A0 × B2) ∪ (A2 × B0) is dominated by A2 × B2,
A1 × B0 is dominated by A1 × B2 and
A0 × B1 is dominated by A2 × B1.
f is a Roman dominating function on G � H.
igyero (EPS-UCA) GRAPHS 19 / 24
Strong product graphs
Roman domination
f1 = (A0,A1,A2), γR(G )-function. f2 = (B0,B1,B2), γR(H)-function.Then,
γR(G � H) ≤ γR(G )γR(H)− 2|A2||B2|.
Idea of the proof
f on G � H defined as
f (u, v) =
2, (u, v) ∈ (A1 × B2) ∪ (A2 × B1) ∪ (A2 × B2),1, (u, v) ∈ A1 × B1,0, otherwise.
(A0 × B0) ∪ (A0 × B2) ∪ (A2 × B0) is dominated by A2 × B2,
A1 × B0 is dominated by A1 × B2 and
A0 × B1 is dominated by A2 × B1.
f is a Roman dominating function on G � H.
igyero (EPS-UCA) GRAPHS 19 / 24
Strong product graphs
Roman domination
f1 = (A0,A1,A2), γR(G )-function. f2 = (B0,B1,B2), γR(H)-function.Then,
γR(G � H) ≤ γR(G )γR(H)− 2|A2||B2|.
Idea of the proof
f on G � H defined as
f (u, v) =
2, (u, v) ∈ (A1 × B2) ∪ (A2 × B1) ∪ (A2 × B2),1, (u, v) ∈ A1 × B1,0, otherwise.
(A0 × B0) ∪ (A0 × B2) ∪ (A2 × B0) is dominated by A2 × B2,
A1 × B0 is dominated by A1 × B2 and
A0 × B1 is dominated by A2 × B1.
f is a Roman dominating function on G � H.
igyero (EPS-UCA) GRAPHS 19 / 24
Strong product graphs
Roman domination
f1 = (A0,A1,A2), γR(G )-function. f2 = (B0,B1,B2), γR(H)-function.Then,
γR(G � H) ≤ γR(G )γR(H)− 2|A2||B2|.
Idea of the proof
f on G � H defined as
f (u, v) =
2, (u, v) ∈ (A1 × B2) ∪ (A2 × B1) ∪ (A2 × B2),1, (u, v) ∈ A1 × B1,0, otherwise.
(A0 × B0) ∪ (A0 × B2) ∪ (A2 × B0) is dominated by A2 × B2,
A1 × B0 is dominated by A1 × B2 and
A0 × B1 is dominated by A2 × B1.
f is a Roman dominating function on G � H.
igyero (EPS-UCA) GRAPHS 19 / 24
Strong product graphs
Roman domination
f1 = (A0,A1,A2), γR(G )-function. f2 = (B0,B1,B2), γR(H)-function.Then,
γR(G � H) ≤ γR(G )γR(H)− 2|A2||B2|.
Idea of the proof
f on G � H defined as
f (u, v) =
2, (u, v) ∈ (A1 × B2) ∪ (A2 × B1) ∪ (A2 × B2),1, (u, v) ∈ A1 × B1,0, otherwise.
(A0 × B0) ∪ (A0 × B2) ∪ (A2 × B0) is dominated by A2 × B2,
A1 × B0 is dominated by A1 × B2 and
A0 × B1 is dominated by A2 × B1.
f is a Roman dominating function on G � H.
igyero (EPS-UCA) GRAPHS 19 / 24
Strong product graphs
Roman domination
f1 = (A0,A1,A2), γR(G )-function. f2 = (B0,B1,B2), γR(H)-function.Then,
γR(G � H) ≤ γR(G )γR(H)− 2|A2||B2|.
Idea of the proof
f on G � H defined as
f (u, v) =
2, (u, v) ∈ (A1 × B2) ∪ (A2 × B1) ∪ (A2 × B2),1, (u, v) ∈ A1 × B1,0, otherwise.
(A0 × B0) ∪ (A0 × B2) ∪ (A2 × B0) is dominated by A2 × B2,
A1 × B0 is dominated by A1 × B2 and
A0 × B1 is dominated by A2 × B1.
f is a Roman dominating function on G � H.
igyero (EPS-UCA) GRAPHS 19 / 24
Rooted product graphs
Index
1 Introduction
2 Cartesian product graphs
3 Strong product graphs
4 Rooted product graphs
igyero (EPS-UCA) GRAPHS 20 / 24
Rooted product graphs
Domination
G , graph of order n ≥ 2. H, graph with root v and at least twovertices. If v does not belong to any γ(H)-set or v belongs to everyγ(H)-set, then
γ(G ◦ H) = nγ(H).
G , graph of order n ≥ 2. Then for any graph H with root v and atleast two vertices,
γ(G ◦ H) ∈ {nγ(H), n(γ(H)− 1) + γ(G )}.
igyero (EPS-UCA) GRAPHS 21 / 24
Rooted product graphs
Domination
G , graph of order n ≥ 2. H, graph with root v and at least twovertices. If v does not belong to any γ(H)-set or v belongs to everyγ(H)-set, then
γ(G ◦ H) = nγ(H).
G , graph of order n ≥ 2. Then for any graph H with root v and atleast two vertices,
γ(G ◦ H) ∈ {nγ(H), n(γ(H)− 1) + γ(G )}.
igyero (EPS-UCA) GRAPHS 21 / 24
Rooted product graphs
Roman domination
G , graph of order n ≥ 2. Then for any graph H with root v and atleast two vertices,
n(γR(H)− 1) + γ(G ) ≤ γR(G ◦ H) ≤ nγR(H).
Tightness of the bounds
If for every γR(H)-function f = (B0,B1,B2) is satisfied thatf (v) = 0, then
γR(G ◦ H) = nγR(H).
If there exist two γR(H)-functions h = (B0,B1,B2) andh′ = (B ′
0,B′1,B
′2) such that h(v) = 1 and h′(v) = 2, then
γR(G ◦ H) = n(γR(H)− 1) + γ(G ).
igyero (EPS-UCA) GRAPHS 22 / 24
Rooted product graphs
Roman domination
G , graph of order n ≥ 2. Then for any graph H with root v and atleast two vertices,
n(γR(H)− 1) + γ(G ) ≤ γR(G ◦ H) ≤ nγR(H).
Tightness of the bounds
If for every γR(H)-function f = (B0,B1,B2) is satisfied thatf (v) = 0, then
γR(G ◦ H) = nγR(H).
If there exist two γR(H)-functions h = (B0,B1,B2) andh′ = (B ′
0,B′1,B
′2) such that h(v) = 1 and h′(v) = 2, then
γR(G ◦ H) = n(γR(H)− 1) + γ(G ).
igyero (EPS-UCA) GRAPHS 22 / 24
Rooted product graphs
Roman domination
G , graph of order n ≥ 2. Then for any graph H with root v and atleast two vertices,
n(γR(H)− 1) + γ(G ) ≤ γR(G ◦ H) ≤ nγR(H).
Tightness of the bounds
If for every γR(H)-function f = (B0,B1,B2) is satisfied thatf (v) = 0, then
γR(G ◦ H) = nγR(H).
If there exist two γR(H)-functions h = (B0,B1,B2) andh′ = (B ′
0,B′1,B
′2) such that h(v) = 1 and h′(v) = 2, then
γR(G ◦ H) = n(γR(H)− 1) + γ(G ).
igyero (EPS-UCA) GRAPHS 22 / 24
Rooted product graphs
G , graph of order n ≥ 2 and H, graph with root v and at least twovertices.
If for every γR(H)-function f is satisfied that f (v) = 1, then
γR(G ◦ H) = n(γR(H)− 1) + γR(G ).
igyero (EPS-UCA) GRAPHS 23 / 24
Rooted product graphs
G , graph of order n ≥ 2 and H, graph with root v and at least twovertices.
If for every γR(H)-function f is satisfied that f (v) = 1, then
γR(G ◦ H) = n(γR(H)− 1) + γR(G ).
igyero (EPS-UCA) GRAPHS 23 / 24
Thanks
THANKS!!!
igyero (EPS-UCA) GRAPHS 24 / 24