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research
Trees with Maximum p-Reinforcement Number
Authors
You Lu
Jun-Ming Xu
Publication date
25 November 2012
Publisher
View
on
arXiv
Abstract
Let
G
=
(
V
,
E
)
G=(V,E)
G
=
(
V
,
E
)
be a graph and
p
p
p
a positive integer. The
p
p
p
-domination number
\g_p(G)
is the minimum cardinality of a set
D
β
V
D\subseteq V
D
β
V
with
β£
N
G
(
x
)
β©
D
β£
β₯
p
|N_G(x)\cap D|\geq p
β£
N
G
β
(
x
)
β©
D
β£
β₯
p
for all
x
β
V
β
D
x\in V\setminus D
x
β
V
β
D
. The
p
p
p
-reinforcement number
r
p
(
G
)
r_p(G)
r
p
β
(
G
)
is the smallest number of edges whose addition to
G
G
G
results in a graph
G
β²
G'
G
β²
with
\g_p(G')<\g_p(G)
. Recently, it was proved by Lu et al. that
r
p
(
T
)
β€
p
+
1
r_p(T)\leq p+1
r
p
β
(
T
)
β€
p
+
1
for a tree
T
T
T
and
p
β₯
2
p\geq 2
p
β₯
2
. In this paper, we characterize all trees attaining this upper bound for
p
β₯
3
p\geq 3
p
β₯
3
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Last time updated on 30/10/2017