Applied Mathematical Sciences, Vol. 6, 2012, no. 87, 4345 - 4352
Domination on Kronecker Product of Pn
Thanin Sitthiwirattham1
Department of Mathematics, Faculty of Applied Science,
King Mongkut’s University of Technology North Bangkok,
Bangkok 10800, Thailand
Centre of Excellence in Mathematics, CHE, Sri Ayutthaya Road,
Bangkok 10400, Thailand
Abstract
Let γ(G) and γ (G) be the domination number and edge domination
number, respectively. The Kronecker Product G1 ⊗ G2 of graph of G1
and G2 has vertex set V (G1 ⊗ G2 ) = V (G1 ) × V (G2 ) and edge set
E(G1 ⊗ G2 ) = {(u1 v1 )(u2 v2 )|u1 u2 ∈ E(G1 ) and v1 v2 ∈ E(G2 )}. In this
paper, let G is a simple graph with order m, we prove that,
γ(Pn ⊗ G) = min nγ(G), m n3 and
γ (Pn ⊗ G) =
min nγ (G), m n−1
,
n is even,
3
n−1
min nγ (G) + γ (Hn [E]), m 3 , n is odd.
Mathematics Subject Classification: 05C69, 05C70, 05C76
Keywords: Kronecker Product, domination number, edge domination
number
1
Introduction
In this paper, graphs must be simple graphs which can be the trivial graph.
Let G1 and G2 be graphs. The Kronecker product of graph G1 and G2 , denote
by G1 ⊗ G2 , is a graph with V (G1 ⊗ G2 ) = V (G1 ) × V (G2 ) and E(G1 ⊗ G2 ) =
{(u1 v1 )(u2v2 )|u1 u2 ∈ E(G1 ) and v1 v2 ∈ E(G2 )}.
In [1], there are some properties about Kronecker product of graph. We
recall these here.
1
[email protected]
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T. Sitthiwirattham
Proposition 1.1. Let H = G1 ⊗ G2 = (V (H), E(H)) then
(i) |V (H)| = |V (G1 )||V (G2 )|
(ii) |E(H)| = 2|E(G1 )||E(G2 )|
(iii) for every (u, v) ∈ V (H), dH ((u, v)) = dG1 (u)dG2 (v).
Theorem 1.2. Let G1 and G2 be connected graphs, The graph H = G1 ⊗G2
is connected if and only if G1 or G2 contains an odd cycle.
Theorem 1.3. Let G1 and G2 be connected graphs with no odd cycle then
G1 ⊗ G2 has exactly two connected components.
Next we get that general form of graph of Kronecker Product of Pn and a
simple graph.
Proposition 1.4. Let G be a connected graph of order m, the graph of
Pn ⊗ G
is
n−1
Hi
i=1
where V (Hi )=Si ∪ Si+1 for i = 1, 2, ..., n − 1; Si = {(i, 1), (i, 2), ..., (i, m)};
E(Hi ) = {(i, u)(i + 1, v)/uv ∈ E(G)}. Moreover, if G has no odd cycle then
each Hi has exactly two connected components isomorphic to G.
Next, we give the definitions about some graph parameters. A dominating
set (or domset) of graph G is a subset D of the vertex set V of G such that
each vertex of V − D is adjacent to at least one vertex of D. The minimum
cardinality of a dominating set of a graph G is called the domination number
of G, denote by γ(G).
A subset T of the edge set E of G is said to be an edge dominating set
of graph G, if each edge of G either is in T , or adjacent to an edge of T .
The minimum cardinality of an edge dominating set of G is called the edge
domination number of G, denoted by γ (G).
By definitions of domination number and edge domination number, clearly
.
that γ(Pn ) = n3 and γ (Pn ) = n−1
3
2
Domination number of the graph of Pn⊗G
We begin this section by giving the theorem 2.1, that shows a character of
minimum dominating set.
Domination on Kronecker product of Pn
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Theorem 2.1. [4] A dominating set D is a minimum dominating set if and
only if for each vertex u ∈ D, one of following two conditions holds:
(a) N(u) ⊆ V − D
(b) there exists a vertex v ∈ V − D for which N(v) ∩ D = {u}.
Next, we giving the lemma 2.2 which show character of domination number
for each Hi .
Lemma 2.2. Let Pn ⊗ G =
n−1
Hi . For each Hi , then γ(Hi ) = 2γ(G).
i=1
Proof. Suppose G has no odd cycle, by proposition 1.4 we get Hi =2G . So
γ(Hi ) = 2γ(G).
If G has odd cycle, for each Hi , vertex (ui , v) ∈ Si and (ui+1 , v) ∈ Si+1
n−1
have dHi ((ui , v)) = dHi (ui+1 , v)) = dG (v). Let
Hi = Pn ⊗ (G − e) when
i=1
e is an edge in odd cycle, D be the minimum dominating set of G . We get
Hi = 2(G − e) then
⎧
2[γ(G) + 1], if e = xy then x ∈ D is in dominating
⎪
⎪
⎪
⎨
set of G, y ∈ D and is not adjecent with
γ(Hi ) = 2γ(G − e) =
⎪
vertex z ∈ D
⎪
⎪
⎩
2γ(G),
otherwise.
When we add e comeback, in the case γ(G − e) = γ(G) + 1 be not
impossible because the end vertices of edge e are in dominating set of G − e,
so γ(Hi ) = γ(Hi ) − 1. Hence γ(Hi ) = 2γ(G). Similarly, γ(Hn ) = 2γ(G).
Example
Figure 1: The graph of P4 , G1 , G2 and G3
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T. Sitthiwirattham
Figure 2: The graph of P4 ⊗ G1
Figure 3: The graph of P4 ⊗ G2 and P4 ⊗ G3
Next, we establish theorem 2.3 for a minimum dominating number of Pn ⊗G
Theorem 2.3. Let G be connected graph order m, then γ(Pn ⊗G) = min{nγ(G), m n3 }.
Proof. Let V (Pn ) = {ui , i = 1, 2, ..., n}, V (G) = {vi , i = 1, 2, ..., m}, Si =
{(vi , uj )
∈ V (Pn ⊗ G)/j = 1, 2, ..., m}, i = 1, 2, ..., n and since γ(Pn ) = n3 . Assume
that the minimum dominating set of Pn , G be
{u2 , u5, ..., u3p−1 }
, n = 3p − 1 or 3p
, D2 = {v1 , v2 , ..., vk },
D1 =
{u2 , u5, ..., u3p−1 , u3p+1 } , n = 3p + 1
respectively.
For each Hi , by lemma 2.2 we have γ(Hi ) = 2γ(G). Since Pn ⊗G is
n−1
Hi
i=1
which is every Hi and Hi+1 have γ(G) common vertices in their dominating
set, then γ(Pn ⊗ G) ≤ nγ(G).
Domination on Kronecker product of Pn
4349
In the author hand, we get a dominating set of Pn ⊗ G be
{S2 , S5 , ..., S3p−1 }
, n = 3p − 1 or 3p
n
, then γ(Pn ⊗ G) ≤ m 3
{S2 , S5 , ..., S3p−1 , S3p+1 } , n = 3p + 1
Hence γ(Pn ⊗ G) ≤ min{nγ(G), m n3 }.
Figure 4: The region of D, S when n = 3k − 1; k = 1, 2, ...
Suppose that γ(Pn ⊗ G) <min{nγ(G), m n3 }, then there exists
uvj (orui v) ∈ V (Pn ⊗ G) − D(orS);
2, 5, .., 3k − 1
, n is 3k − 1 or 3k
j = k + 1, k + 2, ..., m; i =
2, 5, ..., 3k − 1, 3k + 1 , n is 3k + 1
which is not adjacent with another vertices in D (or S) , D = {uvk /vk ∈ D2 }
and S = {uh v/uh ∈ D1 }. It is not true, because for every uvj (oruiv) adjacent
with vertices in D (or S) .
Hence γ(Pn ⊗ G) = min{nγ(G), m n3 }.
3
Edge Domination number of the graph of
Pn⊗G
We begin this section by giving the lemma 3.1 that shows character of edge
domination set for each Hi .
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T. Sitthiwirattham
Lemma 3.1. Let Pn ⊗ G =
n−1
Hi . For each Hi then γ (Hi ) = 2γ (G)
i=1
Proof. Suppose G has no odd cycle, by proposition 1.4, we get Hi =2G . So
γ (Hi) = 2γ (G).
If G has odd cycle, for each (ui , v) ∈ Si , (ui+1 , v) ∈ Si+1 in V (Hi ) have
n−1
dHi ((ui , v)) = dHi (ui+1 , v). Let
Hi = Pn ⊗ (G − e) when e is an edge
i=1
in odd cycle, D be the minimum edge domination set of G. Clearly that
γ (Hi) = 2γ (G − e) = 2γ (G).
Next, we establish theorem 3.2. for a minimum edge covering number of
Pn ⊗ G.
Theorem 3.2. Let G be connected graph order m, then
n−1
min
nγ
(G),
m
,
n is even,
3
γ (Pn ⊗ G) =
n−1
min nγ (G) + γ (Hn [E]), m 3 , n is odd.
where E = {e ∈ Hn | e is not adjacent with edges in the minimum dominating
set of Hn }
Proof. Let V (Pn ) = {ui , i = 1, 2, ..., n}, V (G) = {vj , j = 1, 2, ..., m}, Si =
.
{(ui , vj ) ∈ V (Pn ⊗G)/j = 1, 2, ..., m}, i = 1, 2, ..., n and since γ (Pn ) = n−1
3
Assume that the minimum edge domination set of Pn , G be
{u2 u3 , u5 u6, ..., u3p−2 u3p−1 } where n = 3p − 1
, D2 , respectively.
D1 =
{u2 u3 , u5 u6, ..., u3p−1 u3p }
where n = 3p or 3p + 1.
Figure 5: The edge domination set of Pn ⊗ G where n is even
Domination on Kronecker product of Pn
n−1
4351
For each Hi , by lemma 3.1 we have γ (Hi ) = 2γ (G). Since Pn ⊗ G is
Hi which is every Hi for i is even, all edges in Hi be adjacent with edges
i=1
in the edge domination set of Hi−1 or Hi+1 , then γ (Pn ⊗ G) ≤ nγ (G) where
n is even.
In the author hand, we get another edges domination set of Pn ⊗ G be
K=
E(H2 ) ∪ E(H5 ) ∪ ... ∪ E(H3p−2 )
, n = 3p − 1
,
E(H2 ) ∪ E(H5 ) ∪ ... ∪ E(H3 p − 1) , n = 3p or n = 3p + 1
. Hence γ (Pn ⊗ G) ≤ min{nγ (G), m n−1
}.
then γ (Pn ⊗ G) ≤ m n−1
3
3
Figure 6: The edge domination set of Pn ⊗ G where n = 3p
If nγ (G) < m n−1
, suppose that γ (Pn ⊗ G) < nγ (G), for every Hi and
3
Hi+1 , there exists edge in Hi+1 which is not adjacent with edge in the edge
domination set of Hi . That is not true.
If nγ (G) > m n−1
, suppose that γ (Pn ⊗ G) < m n−1
, then there ex3
3
ists edge such that is not in K, which is not adjacent with edge in the edge
domination set of Hi . That is not true.
} where n is even.
Hence γ (Pn ⊗ G) = min{nγ (G), m n−1
3
Similarly, in case n is odd, we get γ (Pn ⊗ G) ≤ nγ (G) + γ (Hn [E]) where
E = {e ∈ Hn | e is not adjacent with edges in the minimum dominating set of
Hn }.
} where n is odd. Hence γ (Pn ⊗ G) = min{nγ (G) + γ (Hn [E]), m n−1
3
ACKNOWLEDGEMENTS. This research is supported by King Mongkut’s
University of Technology North Bangkok, Thailand.
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T. Sitthiwirattham
Figure 7: The edge domination set of Pn ⊗ G where n is odd.
References
[1] Z.A. Bottreou, Y.Metivier, Some remarks on the Kronecker product of
graph, Inform. Process, Lett. 86 (1998), 279-286.
[2] D.B.West, Introduction to Graph Theory, Prentice-Hall.,2001.
[3] P.M. Welchsel, The Kronecker product of graphs, Proc. Amer. Math,
Soc.8 (1962), 47-52.
[4] O.Ore, Theory of graphs, Amer.Math, Soc.Collog.Publ, 38 (Amer. Math.
Soc., Providence, RI), 1962.
Received: April, 2012