2-Domination number of generalized Petersen graphs
Let G = ( V , E ) be a graph. A subset S ⊆ V is a k - dominating set of G if each vertex in V - S is adjacent to at least k vertices in S . The k - domination number of G is the cardinality of the smallest k -dominating set of G . In this paper, we shall prove that the 2-domination number of general...
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| Veröffentlicht in: | Proceedings of the Indian Academy of Sciences. Mathematical sciences 2018-04, Vol.128 (2), p.1-12, Article 17 |
|---|---|
| Hauptverfasser: | , , |
| Format: | Artikel |
| Sprache: | eng |
| Schlagworte: | |
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| Zusammenfassung: | Let
G
=
(
V
,
E
)
be a graph. A subset
S
⊆
V
is a
k
-
dominating
set of
G
if each vertex in
V
-
S
is adjacent to at least
k
vertices in
S
. The
k
-
domination number
of
G
is the cardinality of the smallest
k
-dominating set of
G
. In this paper, we shall prove that the 2-domination number of generalized Petersen graphs
P
(
5
k
+
1
,
2
)
and
P
(
5
k
+
2
,
2
)
, for
k
>
0
, is
4
k
+
2
and
4
k
+
3
, respectively. This proves two conjectures due to Cheng (Ph.D. thesis, National Chiao Tung University,
2013
). Moreover, we determine the exact 2-domination number of generalized Petersen graphs
P
(2
k
,
k
) and
P
(
5
k
+
4
,
3
)
. Furthermore, we give a good lower and upper bounds on the 2-domination number of generalized Petersen graphs
P
(
5
k
+
1
,
3
)
,
P
(
5
k
+
2
,
3
)
and
P
(
5
k
+
3
,
3
)
. |
|---|---|
| ISSN: | 0253-4142 0973-7685 |
| DOI: | 10.1007/s12044-018-0395-2 |