Nordhaus–Gaddum-Type Results on the Connected Edge Domination Number
A connected edge dominating set of a connected graph G = ( V , E ) is a subset X of E such that the edge-induced subgraph G [ X ] is connected and each e ∈ E ( G ) \ X has at least one neighbor in X . The connected edge domination number γ c ′ ( G ) of G is the minimum cardinality of a connected edg...
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| Published in | Bulletin of the Malaysian Mathematical Sciences Society Vol. 47; no. 2 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Singapore
Springer Nature Singapore
01.03.2024
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
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| Summary: | A
connected edge dominating set
of a connected graph
G
=
(
V
,
E
)
is a subset
X
of
E
such that the edge-induced subgraph
G
[
X
] is connected and each
e
∈
E
(
G
)
\
X
has at least one neighbor in
X
. The
connected edge domination number
γ
c
′
(
G
)
of
G
is the minimum cardinality of a connected edge dominating set of
G
. An edge dominating set
X
of a graph
G
is called a 2
-edge-connected edge dominating set
if
G
[
X
] is 2-edge-connected. The 2
-edge-connected edge dominating number
γ
2
e
c
′
(
G
)
is the minimum size of a 2-edge-connected edge dominating set of
G
. In this paper, we obtain the sharp lower bounds for
γ
c
′
(
G
)
+
γ
c
′
(
G
¯
)
and
γ
c
′
(
G
)
·
γ
c
′
(
G
¯
)
. Moreover, we characterize the classes of graphs attaining the lower bounds and study the relationship between
γ
c
′
(
G
)
and several other parameters, such as independent number and vertex cover number. In addition, we show that
3
≤
γ
2
e
c
′
(
G
)
≤
⌊
3
2
(
n
-
1
)
⌋
if
G
is 2-edge-connected. We also obtain the upper and lower bounds for
γ
2
e
c
′
(
G
)
+
γ
2
e
c
′
(
G
¯
)
and
γ
2
e
c
′
(
G
)
·
γ
2
e
c
′
(
G
¯
)
and characterize the classes of graphs attaining the lower bounds. |
|---|---|
| ISSN: | 0126-6705 2180-4206 |
| DOI: | 10.1007/s40840-023-01649-1 |