Abstract
AbstractA wide range of parameters of domination in graphs can be defined and studied through a common approach that was recently introduced in [https://doi.org/10.26493/1855-3974.2318.fb9] under the name of w-domination, where $$w=(w_0,w_1, \dots ,w_l)$$
w
=
(
w
0
,
w
1
,
⋯
,
w
l
)
is a vector of non-negative integers such that $$ w_0\ge 1$$
w
0
≥
1
. Given a graph G, a function $$f: V(G)\longrightarrow \{0,1,\dots ,l\}$$
f
:
V
(
G
)
⟶
{
0
,
1
,
⋯
,
l
}
is said to be a w-dominating function if $$\sum _{u\in N(v)}f(u)\ge w_i$$
∑
u
∈
N
(
v
)
f
(
u
)
≥
w
i
for every vertex v with $$f(v)=i$$
f
(
v
)
=
i
, where N(v) denotes the open neighbourhood of $$v\in V(G)$$
v
∈
V
(
G
)
. The weight of f is defined to be $$\omega (f)=\sum _{v\in V(G)} f(v)$$
ω
(
f
)
=
∑
v
∈
V
(
G
)
f
(
v
)
, while the w-domination number of G, denoted by $$\gamma _{w}(G)$$
γ
w
(
G
)
, is defined as the minimum weight among all w-dominating functions on G. A wide range of well-known domination parameters can be defined and studied through this approach. For instance, among others, the vector $$w=(1,0)$$
w
=
(
1
,
0
)
corresponds to the case of standard domination, $$w=(2,1)$$
w
=
(
2
,
1
)
corresponds to double domination, $$w=(2,0,0)$$
w
=
(
2
,
0
,
0
)
corresponds to Italian domination, $$w=(2,0,1)$$
w
=
(
2
,
0
,
1
)
corresponds to quasi-total Italian domination, $$w=(2,1,1)$$
w
=
(
2
,
1
,
1
)
corresponds to total Italian domination, $$w=(2,2,2)$$
w
=
(
2
,
2
,
2
)
corresponds to total $$\{2\}$$
{
2
}
-domination, while $$w=(k,k-1,\dots ,1,0)$$
w
=
(
k
,
k
-
1
,
⋯
,
1
,
0
)
corresponds to $$\{k\}$$
{
k
}
-domination. In this paper, we show that several domination parameters of lexicographic product graphs $$G\circ H$$
G
∘
H
are equal to $$\gamma _{w}(G)$$
γ
w
(
G
)
for some vector $$w\in \{2\}\times \{0,1,2\}^{l}$$
w
∈
{
2
}
×
{
0
,
1
,
2
}
l
and $$l\in \{2,3\}$$
l
∈
{
2
,
3
}
. The decision on whether the equality holds for a specific vector w will depend on the value of some domination parameters of H. In particular, we focus on quasi-total Italian domination, total Italian domination, 2-domination, double domination, total $$\{2\}$$
{
2
}
-domination, and double total domination of lexicographic product graphs.
Funder
Universitat Rovira i Virgili
Publisher
Springer Science and Business Media LLC