Loading…
Secure Italian domination in graphs
An Italian dominating function (IDF) on a graph G is a function f : V ( G ) → { 0 , 1 , 2 } such that for every vertex v with f ( v ) = 0 , the total weight of f assigned to the neighbours of v is at least two, i.e., ∑ u ∈ N G ( v ) f ( u ) ≥ 2 . For any function f : V ( G ) → { 0 , 1 , 2 } and any...
Saved in:
| Published in: | Journal of combinatorial optimization 2021, Vol.41 (1), p.56-72 |
|---|---|
| Main Authors: | , , |
| Format: | Article |
| Language: | English |
| Subjects: | |
| Citations: | Items that this one cites Items that cite this one |
| Online Access: | Get full text |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | An Italian dominating function (IDF) on a graph
G
is a function
f
:
V
(
G
)
→
{
0
,
1
,
2
}
such that for every vertex
v
with
f
(
v
)
=
0
, the total weight of
f
assigned to the neighbours of
v
is at least two, i.e.,
∑
u
∈
N
G
(
v
)
f
(
u
)
≥
2
. For any function
f
:
V
(
G
)
→
{
0
,
1
,
2
}
and any pair of adjacent vertices with
f
(
v
)
=
0
and
u
with
f
(
u
)
>
0
, the function
f
u
→
v
is defined by
f
u
→
v
(
v
)
=
1
,
f
u
→
v
(
u
)
=
f
(
u
)
-
1
and
f
u
→
v
(
x
)
=
f
(
x
)
whenever
x
∈
V
(
G
)
\
{
u
,
v
}
. A secure Italian dominating function on a graph
G
is defined as an IDF
f
which satisfies that for every vertex
v
with
f
(
v
)
=
0
, there exists a neighbour
u
with
f
(
u
)
>
0
such that
f
u
→
v
is an IDF. The weight of
f
is
ω
(
f
)
=
∑
v
∈
V
(
G
)
f
(
v
)
. The minimum weight among all secure Italian dominating functions on
G
is the secure Italian domination number of
G
. This paper is devoted to initiating the study of the secure Italian domination number of a graph. In particular, we prove that the problem of finding this parameter is NP-hard and we obtain general bounds on it. Moreover, for certain classes of graphs, we obtain closed formulas for this novel parameter. |
|---|---|
| ISSN: | 1382-6905 1573-2886 |
| DOI: | 10.1007/s10878-020-00658-1 |