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research
Locating-total dominating sets in twin-free graphs: a conjecture
Authors
Florent Foucaud
Michael A. Henning
Publication date
1 January 2016
Publisher
View
on
arXiv
Abstract
A total dominating set of a graph
G
G
G
is a set
D
D
D
of vertices of
G
G
G
such that every vertex of
G
G
G
has a neighbor in
D
D
D
. A locating-total dominating set of
G
G
G
is a total dominating set
D
D
D
of
G
G
G
with the additional property that every two distinct vertices outside
D
D
D
have distinct neighbors in
D
D
D
; that is, for distinct vertices
u
u
u
and
v
v
v
outside
D
D
D
,
N
(
u
)
∩
D
â‰
N
(
v
)
∩
D
N(u) \cap D \ne N(v) \cap D
N
(
u
)
∩
D
î€
=
N
(
v
)
∩
D
where
N
(
u
)
N(u)
N
(
u
)
denotes the open neighborhood of
u
u
u
. A graph is twin-free if every two distinct vertices have distinct open and closed neighborhoods. The location-total domination number of
G
G
G
, denoted
L
T
(
G
)
LT(G)
L
T
(
G
)
, is the minimum cardinality of a locating-total dominating set in
G
G
G
. It is well-known that every connected graph of order
n
≥
3
n \geq 3
n
≥
3
has a total dominating set of size at most
2
3
n
\frac{2}{3}n
3
2
​
n
. We conjecture that if
G
G
G
is a twin-free graph of order
n
n
n
with no isolated vertex, then
L
T
(
G
)
≤
2
3
n
LT(G) \leq \frac{2}{3}n
L
T
(
G
)
≤
3
2
​
n
. We prove the conjecture for graphs without
4
4
4
-cycles as a subgraph. We also prove that if
G
G
G
is a twin-free graph of order
n
n
n
, then
L
T
(
G
)
≤
3
4
n
LT(G) \le \frac{3}{4}n
L
T
(
G
)
≤
4
3
​
n
.Comment: 18 pages, 1 figur
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Last time updated on 08/01/2021