Related papers: Nonblocker in H-minor free graphs: kernelization m…
The orthogonality dimension of a graph over $\mathbb{R}$ is the smallest integer $d$ for which one can assign to every vertex a nonzero vector in $\mathbb{R}^d$ such that every two adjacent vertices receive orthogonal vectors. For an…
Power domination in graphs emerged from the problem of monitoring an electrical system by placing as few measurement devices in the system as possible. It corresponds to a variant of domination that includes the possibility of propagation.…
Given a positive integer $k$, a $k$-dominating set in a graph $G$ is a set of vertices such that every vertex not in the set has at least $k$ neighbors in the set. A total $k$-dominating set, also known as a $k$-tuple total dominating set,…
A set $S\subseteq V$ of a graph $G=(V,E)$ is a dominating set if each vertex has a neighbor in $S$ or belongs to $S$. Dominating Set is the problem of deciding, given a graph $G$ and an integer $k\geq 1$, if $G$ has a dominating set of size…
We explore a reconfiguration version of the dominating set problem, where a dominating set in a graph $G$ is a set $S$ of vertices such that each vertex is either in $S$ or has a neighbour in $S$. In a reconfiguration problem, the goal is…
Tracking of moving objects is crucial to security systems and networks. Given a graph $G$, terminal vertices $s$ and $t$, and an integer $k$, the \textsc{Tracking Paths} problem asks whether there exists at most $k$ vertices, which if…
For a graph $H$, a graph $G$ is an $H$-graph if it is an intersection graph of connected subgraphs of some subdivision of $H$. $H$-graphs naturally generalize several important graph classes like interval or circular-arc graph. This class…
We show that every planar triangulation on $n>10$ vertices has a dominating set of size $n/7=n/3.5$. This approaches the $n/4$ bound conjectured by Matheson and Tarjan [MT'96], and improves significantly on the previous best bound of…
Many graph problems were first shown to be fixed-parameter tractable using the results of Robertson and Seymour on graph minors. We show that the combination of finite, computable, obstruction sets and efficient order tests is not just one…
A locating-dominating set $D$ of a graph $G$ is a dominating set of $G$ where each vertex not in $D$ has a unique neighborhood in $D$, and the Locating-Dominating Set problem asks if $G$ contains such a dominating set of bounded size. This…
The problems of determining the minimum-sized \emph{identifying}, \emph{locating-dominating} and \emph{open locating-dominating codes} of an input graph are special search problems that are challenging from both theoretical and…
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…
The composition technique is a popular method for excluding polynomial-size problem kernels for NP-hard parameterized problems. We present a new technique exploiting triangle-based fractal structures for extending the range of applicability…
A stable cutset in a graph $G$ is a set $S\subseteq V(G)$ such that vertices of $S$ are pairwise non-adjacent and such that $G-S$ is disconnected, i.e., it is both stable (or independent) set and a cutset (or separator). Unlike general…
The F-Minor-Free Deletion problem asks, for a fixed set F and an input consisting of a graph G and integer k, whether k vertices can be removed from G such that the resulting graph does not contain any member of F as a minor. This paper…
For planar graphs, we consider the problems of \emph{list edge coloring} and \emph{list total coloring}. Edge coloring is the problem of coloring the edges while ensuring that two edges that are adjacent receive different colors. Total…
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…
{\em Partial domination problem} is a generalization of the {\em minimum dominating set problem} on graphs. Here, instead of dominating all the nodes, one asks to dominate at least a fraction of the nodes of the given graph by choosing a…
In the Maximum Minimal Vertex Cover (MMVC) problem, we are given a graph $G$ and a positive integer $k$, and the objective is to decide whether $G$ contains a minimal vertex cover of size at least $k$. Motivated by the kernelization of MMVC…
For a finite collection of connected graphs $\mathcal{F}$, the $\mathcal{F}$-MINOR-DELETION problem consists in, given a graph $G$ and an integer $\ell$, deciding whether $G$ contains a vertex set of size at most $\ell$ whose removal…