Affiliation:
1. The Institute of Mathematical Sciences, Chennai, India
2. RWTH Aachen University, Aachen, Germany
Abstract
We show that for every fixed
j
≥
i
≥ 1, the
k
-D
ominating
S
et
problem restricted to graphs that do not have
K
ij
(the complete bipartite graph on (
i
+
j
) vertices, where the two parts have
i
and
j
vertices, respectively) as a subgraph is fixed parameter tractable (FPT) and has a polynomial kernel. We describe a polynomial-time algorithm that, given a
K
i,j
-free graph
G
and a nonnegative integer
k
, constructs a graph
H
(the “kernel”) and an integer
k
' such that (1)
G
has a dominating set of size at most
k
if and only if
H
has a dominating set of size at most
k
', (2)
H
has
O
((
j
+ 1)
i
+ 1
k
i
2
) vertices, and (3)
k
' =
O
((
j
+ 1)
i
+ 1
k
i
2
).
Since
d
-degenerate graphs do not have
K
d+1,d+1
as a subgraph, this immediately yields a polynomial kernel on
O
((
d
+ 2)
d
+2
k
(
d
+ 1)
2
) vertices for the
k
-D
ominating
S
et
problem on
d
-degenerate graphs, solving an open problem posed by Alon and Gutner [Alon and Gutner 2008; Gutner 2009].
The most general class of graphs for which a polynomial kernel was previously known for
k
-D
ominating
S
et
is the class of
K
h
-topological-minor-free graphs [Gutner 2009]. Graphs of bounded degeneracy are the most general class of graphs for which an FPT algorithm was previously known for this problem.
K
h
-topological-minor-free graphs are
K
i,j
-free for suitable values of
i,j
(but not vice-versa), and so our results show that
k
-D
ominating
S
et
has both FPT algorithms and polynomial kernels in strictly more general classes of graphs.
Using the same techniques, we also obtain an
O
(
jk
i
) vertex-kernel for the
k
-I
ndependent
D
ominating
S
et
problem on
K
i,j
-free graphs.
Publisher
Association for Computing Machinery (ACM)
Subject
Mathematics (miscellaneous)
Cited by
54 articles.
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