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

Abstract. The domination number of a graph G = (V, E) is the minimum size of a dominating set U ⊆ V, which satisfies that every vertex in V \ 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 Kh 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 K3,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 K3,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( √ 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. Key words: H-minor-free graphs, degenerated graphs, dominating set problem, fixed-parameter tractable algorithms, problem kernel

{Linear Kernels for kk-Tupel and Liar's Domination in Bounded Genus Graphs}

A set $D\subseteq V$ is called a $k$-tuple dominating set of a graph $G=(V,E)$ if $\left| N_G[v] \cap D \right| \geq k$ for all $v \in V$, where $N_G[v]$ denotes the closed neighborhood of $v$. A set $D \subseteq V$ is called a liar's dominating set of a graph $G=(V,E)$ if (i) $\left| N_G[v] \cap D \right| \geq 2$ for all $v\in V$ and (ii) for every pair of distinct vertices $u, v\in V$, $\left| (N_G[u] \cup N_G[v]) \cap D \right| \geq 3$. Given a graph $G$, the decision versions of $k$-Tuple Domination Problem and the Liar's Domination Problem are to check whether there exists a $k$-tuple dominating set and a liar's dominating set of $G$ of a given cardinality, respectively. These two problems are known to be NP-complete \cite{LiaoChang2003, Slater2009}. In this paper, we study the parameterized complexity of these problems. We show that the $k$-Tuple Domination Problem and the Liar's Domination Problem are $\mathsf{W}[2]$-hard for general graphs but they admit linear kernels for graphs with bounded genus.Title changed from "Parameterized complexity of k-tuple and liar's domination" to "Linear kernels for k-tuple and liar's domination in bounded genus graphs

Kernelization and Sparseness: the case of Dominating Set

We prove that for every positive integer $r$ and for every graph class $\mathcal G$ of bounded expansion, the $r$-Dominating Set problem admits a linear kernel on graphs from $\mathcal G$. Moreover, when $\mathcal G$ is only assumed to be nowhere dense, then we give an almost linear kernel on $\mathcal G$ for the classic Dominating Set problem, i.e., for the case $r=1$. These results generalize a line of previous research on finding linear kernels for Dominating Set and $r$-Dominating Set. However, the approach taken in this work, which is based on the theory of sparse graphs, is radically different and conceptually much simpler than the previous approaches. We complement our findings by showing that for the closely related Connected Dominating Set problem, the existence of such kernelization algorithms is unlikely, even though the problem is known to admit a linear kernel on $H$-topological-minor-free graphs. Also, we prove that for any somewhere dense class $\mathcal G$, there is some $r$ for which $r$-Dominating Set is W[$2$]-hard on $\mathcal G$. Thus, our results fall short of proving a sharp dichotomy for the parameterized complexity of $r$-Dominating Set on subgraph-monotone graph classes: we conjecture that the border of tractability lies exactly between nowhere dense and somewhere dense graph classes.Comment: v2: new author, added results for r-Dominating Sets in bounded expansion graph