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.

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.