Related papers: Kernelization and approximation of distance-$r$ in…

Let $G=(V,E)$ be a simple graph. A set $S\subseteq V$ is independent set of $G$, if no two vertices of $S$ are adjacent. The independence number $\alpha(G)$ is the size of a maximum independent set in the graph. %An independent set with…

Perhaps the best known kernelization result is the kernel of size 335k for the Planar Dominating Set problem by Alber et al. [JACM 2004], later improved to 67k by Chen et al. [SICOMP 2007]. This result means roughly, that the problem of…

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. A set $I_0(G) \subseteq V(G)$ is a vertex independent set if no two vertices in $I_0(G)$ are adjacent in $G$. We study $\alpha_1(G)$, which is the maximum cardinality of a set…

In this article, we study the $d$-distance $m$-tuple ($\ell, r$)-domination problem. Given a simple undirected graph $G=(V, E)$, and positive integers $d, m, \ell$ and $r$, a subset $V' \subseteq V$ is said to be a $d$-distance $m$-tuple…

In this article, we study a generalized version of the maximum independent set and minimum dominating set problems, namely, the maximum $d$-distance independent set problem and the minimum $d$-distance dominating set problem on unit disk…

A {\em $k$-kernel} in a digraph $G$ is a stable set $X$ of vertices such that every vertex of $G$ can be joined from $X$ by a directed path of length at most $k$. We prove three results about $k$-kernels. First, it was conjectured by…

The independent set problem is NP-hard and particularly difficult to solve in large sparse graphs. In this work, we develop an advanced evolutionary algorithm, which incorporates kernelization techniques to compute large independent sets in…

The Vertex Cover problem plays an essential role in the study of polynomial kernelization in parameterized complexity, i.e., the study of provable and efficient preprocessing for NP-hard problems. Motivated by the great variety of positive…

An independent vertex subset $S$ of the directed graph $G$ is a kernel if the set of out-neighbors of $S$ is $V(G)\setminus S$. An independent vertex subset $Q$ of $G$ is a quasi-kernel if the union of the first and second out-neighbors…

A set $D$ of vertices in an isolate-free graph $G$ is a semitotal dominating set of $G$ if $D$ is a dominating set of $G$ and every vertex in $D$ is within distance $2$ from another vertex of $D$.The semitotal domination number of $G$ is…

Let G=(V,E). A set S is independent if no two vertices from S are adjacent. The number d(X)= |X|-|N(X)| is the difference of X, and an independent set A is critical if d(A) = max{d(I):I is an independent set}. Let us recall that ker(G) is…

The notions of bounded expansion and nowhere denseness have been applied very successfully in algorithmic graph theory. We study the corresponding notions of directed bounded expansion and nowhere crownfulness on directed graphs. We show…

Let integers $r\ge 2$ and $d\ge 3$ be fixed. Let ${\cal G}_d$ be the set of graphs with no induced path on $d$ vertices. We study the problem of packing $k$ vertex-disjoint copies of $K_{1,r}$ ($k\ge 2$) into a graph $G$ from parameterized…

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…

A dominating set of a graph $G$ is a set $D \subseteq V(G)$ such that every vertex in $V(G) \setminus D$ is adjacent to at least one vertex in $D$. A set $L\subseteq V(G)$ is a locating set of $G$ if every vertex in $V(G) \setminus L$ has…

We study kernelization (a kind of efficient preprocessing) for NP-hard problems on planar graphs. Our main result is a kernel of size at most 9k vertices for the Planar Maximum Nonseparating Independent Set problem. A direct consequence of…

Let G=(V,E) be a graph. A set S is independent if no two vertices from S are adjacent. The independence number alpha(G) is the cardinality of a maximum independent set, and mu(G) is the size of a maximum matching. The number…

For $k \geq 1$, in a graph $G=(V,E)$, a set of vertices $D$ is a distance $k$-dominating set of $G$, if any vertex in $V\setminus D$ is at distance at most $k$ from some vertex in $D$. The minimum cardinality of a distance $k$-dominating…

An independent dominating set D of a graph G = (V,E) is a subset of vertices such that every vertex in V \ D has at least one neighbor in D and D is an independent set, i.e. no two vertices of D are adjacent in G. Finding a minimum…

In the NP-hard Edge Dominating Set problem (EDS) we are given a graph $G=(V,E)$ and an integer $k$, and need to determine whether there is a set $F\subseteq E$ of at most $k$ edges that are incident with all (other) edges of $G$. It is…