Related papers: Kernelization and Sparseness: the case of Dominati…
The theoretical notions of graph classes with bounded expansion and that are nowhere dense are meant to capture structural sparsity of real world networks that can be used to design efficient algorithms. In the area of sparse graphs, the…
We prove that whenever $G$ is a graph from a nowhere dense graph class $\mathcal{C}$, and $A$ is a subset of vertices of $G$, then the number of subsets of $A$ that are realized as intersections of $A$ with $r$-neighborhoods of vertices of…
For a positive integer $r$, a distance-$r$ independent set in an undirected graph $G$ is a set $I\subseteq V(G)$ of vertices pairwise at distance greater than $r$, while a distance-$r$ dominating set is a set $D\subseteq V(G)$ such that…
A total dominating set of a graph $G=(V,E)$ is a subset $D \subseteq V$ such that every vertex in $V$ is adjacent to some vertex in $D$. Finding a total dominating set of minimum size is NP-hard on planar graphs and W[2]-complete on general…
We give the first linear kernels for the (Connected) Dominating Set problems on H-topological minor free graphs. We prove the existence of polynomial time algorithms that, for a given H-topological-minor-free graph G and a positive integer…
For $\alpha > 1$, an $\alpha$-approximate (bi-)kernel is a polynomial-time algorithm that takes as input an instance $(I, k)$ of a problem $\mathcal{Q}$ and outputs an instance $(I',k')$ (of a problem $\mathcal{Q}'$) of size bounded by a…
Let $\mathcal{F}$ be a family of graphs, and let $p,r$ be nonnegative integers. The \textsc{$(p,r,\mathcal{F})$-Covering} problem asks whether for a graph $G$ and an integer $k$, there exists a set $D$ of at most $k$ vertices in $G$ such…
Meta-theorems for polynomial (linear) kernels have been the subject of intensive research in parameterized complexity. Heretofore, meta-theorems for linear kernels exist on graphs of bounded genus, $H$-minor-free graphs, and…
For $\alpha\colon\mathbb{N}\rightarrow\mathbb{R}$, an $\alpha$-approximate bi-kernel is a polynomial-time algorithm that takes as input an instance $(I, k)$ of a problem $Q$ and outputs an instance $(I',k')$ of a problem $Q'$ of size…
Inspired by the potential of improving tractability via gap- or above-guarantee parametrisations, we investigate the complexity of Dominating Set when given a suitable lower-bound witness. Concretely, we consider being provided with a…
Let $\mathcal{Q}$ be a vertex subset problem on graphs. In a reconfiguration variant of $\mathcal{Q}$ we are given a graph $G$ and two feasible solutions $S_s, S_t\subseteq V(G)$ of $\mathcal{Q}$ with $|S_s|=|S_t|=k$. The problem is to…
Several algorithmic meta-theorems on kernelization have appeared in the last years, starting with the result of Bodlaender et al. [FOCS 2009] on graphs of bounded genus, then generalized by Fomin et al. [SODA 2010] to graphs excluding a…
For a given graph $G = (V, E)$, a subset of the vertices $D\subseteq V$ is called a semitotal dominating set, if $D$ is a dominating set and every vertex $v \in D$ is within distance two to another witness $v' \in D$. We want to find a…
We show that problems which have finite integer index and satisfy a requirement we call treewidth-bounding admit linear kernels on the class of $H$-topological-minor free graphs, for an arbitrary fixed graph $H$. This builds on earlier…
We show that the k-Dominating Set problem is fixed parameter tractable (FPT) and has a polynomial kernel for any class of graphs that exclude K_{i,j} as a subgraph, for any fixed i, j >= 1. This strictly includes every class of graphs for…
In the last years, kernelization with structural parameters has been an active area of research within the field of parameterized complexity. As a relevant example, Gajarsk{\`y} et al. [ESA 2013] proved that every graph problem satisfying a…
Nowhere dense classes of graphs are very general classes of uniformly sparse graphs with several seemingly unrelated characterisations. From an algorithmic perspective, a characterisation of these classes in terms of uniform quasi-wideness,…
We algorithmize the recent structural characterization for claw-free graphs by Chudnovsky and Seymour. Building on this result, we show that Dominating Set on claw-free graphs is (i) fixed-parameter tractable and (ii) even possesses a…
Kernelization studies polynomial-time preprocessing algorithms. Over the last 20 years, the most celebrated positive results of the field have been linear kernels for classical NP-hard graph problems on sparse graph classes. In this paper,…
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…