Component Order Connectivity in Directed Graphs
A directed graph D is semicomplete if for every pair x , y of vertices of D , there is at least one arc between x and y . Thus, a tournament is a semicomplete digraph. In the Directed Component Order Connectivity (DCOC) problem, given a digraph D = ( V , A ) and a pair of natural numbers k and ℓ ,...
Saved in:
| Published in: | Algorithmica Vol. 84; no. 9; pp. 2767 - 2784 |
|---|---|
| Main Authors: | , , , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
New York
Springer US
01.09.2022
Springer Nature B.V |
| Subjects: | |
| ISSN: | 0178-4617, 1432-0541 |
| Online Access: | Get full text |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | A directed graph
D
is semicomplete if for every pair
x
,
y
of vertices of
D
, there is at least one arc between
x
and
y
. Thus, a tournament is a semicomplete digraph. In the Directed Component Order Connectivity (DCOC) problem, given a digraph
D
=
(
V
,
A
)
and a pair of natural numbers
k
and
ℓ
, we are to decide whether there is a subset
X
of
V
of size
k
such that the largest strongly connected component in
D
-
X
has at most
ℓ
vertices. Note that DCOC reduces to the Directed Feedback Vertex Set problem for
ℓ
=
1
.
We study the parameterized complexity of DCOC for general and semicomplete digraphs with the following parameters:
k
,
ℓ
,
ℓ
+
k
and
n
-
ℓ
. In particular, we prove that DCOC with parameter
k
on semicomplete digraphs can be solved in time
O
∗
(
2
16
k
)
but not in time
O
∗
(
2
o
(
k
)
)
unless the Exponential Time Hypothesis (ETH) fails. The upper bound
O
∗
(
2
16
k
)
implies the upper bound
O
∗
(
2
16
(
n
-
ℓ
)
)
for the parameter
n
-
ℓ
.
We complement the latter by showing that there is no algorithm of time complexity
O
∗
(
2
o
(
n
-
ℓ
)
)
unless ETH fails. Finally, we improve (in dependency on
ℓ
) the upper bound of Göke, Marx and Mnich (2019) for the time complexity of DCOC with parameter
ℓ
+
k
on general digraphs from
O
∗
(
2
O
(
k
ℓ
log
(
k
ℓ
)
)
)
to
O
∗
(
2
O
(
k
log
(
k
ℓ
)
)
)
.
Note that Drange, Dregi and van ’t Hof (2016) proved that even for the undirected version of DCOC on split graphs there is no algorithm of running time
O
∗
(
2
o
(
k
log
ℓ
)
)
unless ETH fails and it is a long-standing problem to decide whether Directed Feedback Vertex Set admits an algorithm of time complexity
O
∗
(
2
o
(
k
log
k
)
)
. |
|---|---|
| Bibliography: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0178-4617 1432-0541 |
| DOI: | 10.1007/s00453-022-01004-z |