Total restrained domination in graphs of diameter 2 or 3
For a given connected graph G =( V , E ), a set D tr ⊆ V ( G ) is a total restrained dominating set if it is a dominating set and both 〈 D t r 〉 and 〈 V ( G )− D t r 〉 do not contain isolated vertices. The cardinality of the minimum total restrained dominating set in G is the total restrained domina...
Saved in:
Published in | Mathematical sciences (Karaj, Iran) Vol. 7; no. 1; p. 1 |
---|---|
Main Authors | , , , , |
Format | Journal Article |
Language | English |
Published |
Berlin/Heidelberg
Springer Berlin Heidelberg
01.12.2013
Springer Nature B.V |
Subjects | |
Online Access | Get full text |
Cover
Loading…
Summary: | For a given connected graph
G
=(
V
,
E
), a set
D
tr
⊆
V
(
G
)
is a total restrained dominating set if it is a dominating set and both 〈
D
t
r
〉 and 〈
V
(
G
)−
D
t
r
〉 do not contain isolated vertices. The cardinality of the minimum total restrained dominating set in
G
is the total restrained domination number and is denoted by
γ
t
r
(
G
). In this paper, we continue the study of total restrained domination number of graphs. We first give some results on total restrained domination number of graphs. And then, we characterize all graphs
G
of order
n
for which (1)
γ
t
r
(
G
)=
n
, (2)
γ
(
G
)=1 and
γ
t
r
(
G
)=3, and (3)
γ
t
r
(
G
)=2. Furthermore, we give some bounds on total restrained domination number of graphs with diameter 3. Finally, we present some bounds for total restrained domination number of some planar graphs with diameter 2 and
γ
-set of cardinality 2. |
---|---|
ISSN: | 2008-1359 2251-7456 |
DOI: | 10.1186/2251-7456-7-26 |