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Revision #1 to TR08-066 | 14th February 2009 00:00

#### Polynomial Kernels and Faster Algorithms for the Dominating Set Problem on Graphs with an Excluded Minor

Revision #1
Authors: Noga Alon, Shai Gutner
Accepted on: 14th February 2009 00:00
Keywords:

Abstract:

The domination number of a graph $G=(V,E)$ is the minimum size of
a dominating set $U \subseteq V$, which satisfies that every
vertex in $V \setminus U$ is adjacent to at least one vertex in
$U$. The notion of a problem kernel refers to a polynomial time
algorithm that achieves some provable reduction of the input size.
Given a graph $G$ whose domination number is $k$, the objective is
to design a polynomial time algorithm that produces a graph $G'$
whose size depends only on $k$, and also has domination number
equal to $k$.
Note that the graph $G'$ is constructed without knowing the value
of $k$.
Problem kernels can be used to obtain efficient approximation and
exact algorithms for the domination number, and are also useful in
practical settings.

In this paper, we present the first nontrivial result for the
general case of graphs with an excluded minor, as follows. For every
fixed $h$, given a graph $G$ with $n$ vertices that does not contain
$K_h$ as a topological minor, our $O(n^{3.5}+k^{O(1)})$ time
algorithm constructs a subgraph $G'$ of $G$, such that if the
domination number of $G$ is $k$, then the domination number of $G'$
is also $k$ and $G'$ has at most $k^c$ vertices, where $c$ is a
constant that depends only on $h$. This result is improved for
graphs that do not contain $K_{3,h}$ as a topological minor, using a
simpler algorithm that constructs a subgraph with at most $ck$
vertices, where $c$ is a constant that depends only on $h$.

Our results imply that there is a problem kernel of polynomial size
for graphs with an excluded minor and a linear kernel for graphs
that are $K_{3,h}$-minor-free. The only previous kernel results
known for the dominating set problem are the existence of a linear
kernel for the planar case as well as for graphs of bounded genus.
Using the polynomial kernel construction, we give an $O(n^{3.5} + 2^{O(\sqrt{k})})$ time algorithm for finding a dominating set of
size at most $k$ in an $H$-minor-free graph with $n$ vertices. This
improves the running time of the previously best known algorithm.

### Paper:

TR08-066 | 16th July 2008 00:00

#### Kernels for the Dominating Set Problem on Graphs with an Excluded Minor

TR08-066
Authors: Noga Alon, Shai Gutner
Publication: 20th July 2008 16:09
Keywords:

Abstract:

The domination number of a graph $G=(V,E)$ is the minimum size of
a dominating set $U \subseteq V$, which satisfies that every
vertex in $V \setminus U$ is adjacent to at least one vertex in
$U$. The notion of a problem kernel refers to a polynomial time
algorithm that achieves some provable reduction of the input size.
Given a graph $G$ whose domination number is $k$, the objective is
to design a polynomial time algorithm that produces a graph $G'$
whose size depends only on $k$, and also has domination number
equal to $k$.
Note that the graph $G'$ is constructed without knowing the value
of $k$.
Problem kernels can be used to obtain efficient approximation and
exact algorithms for the domination number, and are also useful in
practical settings.
In this paper, we present the first nontrivial result for the
general case of graphs with an excluded minor, as follows. For
every fixed $h$, given a graph $G$ that does not contain $K_h$ as
a topological minor, our polynomial time algorithm constructs a
subgraph $G'$ of $G$, such that if the domination number of $G$ is
$k$, then the domination number of $G'$ is also $k$ and $G'$ has
at most $k^c$ vertices, where $c$ is a constant that depends only
on $h$. This result is improved for graphs that do not contain
$K_{3,h}$ as a topological minor, using a simpler algorithm that
constructs a subgraph with at most $ck$ vertices, where $c$ is a
constant that depends only on $h$.
Our results imply that there is a problem kernel of polynomial
size for graphs with an excluded minor and a linear kernel for
graphs that are $K_{3,h}$-minor-free. The only previous kernel
results known for the dominating set problem are the existence of
a linear kernel for the planar case as well as for graphs of
bounded genus.

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