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research
3-Factor-criticality in double domination edge critical graphs
Authors
Erfang Shan
Haichao Wang
Yancai Zhao
Publication date
18 August 2014
Publisher
View
on
arXiv
Abstract
A vertex subset
S
S
S
of a graph
G
G
G
is a double dominating set of
G
G
G
if
∣
N
[
v
]
∩
S
∣
≥
2
|N[v]\cap S|\geq 2
∣
N
[
v
]
∩
S
∣
≥
2
for each vertex
v
v
v
of
G
G
G
, where
N
[
v
]
N[v]
N
[
v
]
is the set of the vertex
v
v
v
and vertices adjacent to
v
v
v
. The double domination number of
G
G
G
, denoted by
γ
×
2
(
G
)
\gamma_{\times 2}(G)
γ
×
2
​
(
G
)
, is the cardinality of a smallest double dominating set of
G
G
G
. A graph
G
G
G
is said to be double domination edge critical if
γ
×
2
(
G
+
e
)
<
γ
×
2
(
G
)
\gamma_{\times 2}(G+e)<\gamma_{\times 2}(G)
γ
×
2
​
(
G
+
e
)
<
γ
×
2
​
(
G
)
for any edge
e
∉
E
e \notin E
e
∈
/
E
. A double domination edge critical graph
G
G
G
with
γ
×
2
(
G
)
=
k
\gamma_{\times 2}(G)=k
γ
×
2
​
(
G
)
=
k
is called
k
k
k
-
γ
×
2
(
G
)
\gamma_{\times 2}(G)
γ
×
2
​
(
G
)
-critical. A graph
G
G
G
is
r
r
r
-factor-critical if
G
−
S
G-S
G
−
S
has a perfect matching for each set
S
S
S
of
r
r
r
vertices in
G
G
G
. In this paper we show that
G
G
G
is 3-factor-critical if
G
G
G
is a 3-connected claw-free
4
4
4
-
γ
×
2
(
G
)
\gamma_{\times 2}(G)
γ
×
2
​
(
G
)
-critical graph of odd order with minimum degree at least 4 except a family of graphs.Comment: 14 page
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Last time updated on 30/10/2017