Parity of an odd dominating set
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Date
2022
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
Ankara Üniversitesi
Abstract
For a simple graph
G
with vertex set
V
(
G
)
=
{
v
1
,
.
.
.
,
v
n
}
, we define the closed neighborhood set of a vertex
u
as \\
N
[
u
]
=
{
v
∈
V
(
G
)
|
v
is adjacent to
u
or
v
=
u
}
and the closed neighborhood matrix
N
(
G
)
as the matrix whose
i
th column is the characteristic vector of
N
[
v
i
]
. We say a set
S
is odd dominating if
N
[
u
]
∩
S
is odd for all
u
∈
V
(
G
)
. We prove that the parity of the cardinality of an odd dominating set of
G
is equal to the parity of the rank of
G
, where rank of
G
is defined as the dimension of the column space of
N
(
G
)
. Using this result we prove several corollaries in one of which we obtain a general formula for the nullity of the join of graphs.
Description
Keywords
Lights out, all-ones problem, odd dominating set, parity domination, domination number