Related papers: Quasi-kernels in split graphs

In a digraph, a quasi-kernel is a subset of vertices that is independent and such that every vertex can reach some vertex in that set via a directed path of length at most two. Whereas Chv\'atal and Lov\'asz proved in 1974 that every…

A {\em quasi-kernel} of a digraph $D$ is an independent set $Q\subseteq V(D)$ such that for every vertex $v\in V(D)\backslash Q$, there exists a directed path with one or two arcs from $v$ to a vertex $u\in Q$. In 1974, Chv\'{a}tal and…

Let $D=(V,A)$ be a digraph. A vertex set $K\subseteq V$ is a quasi-kernel of $D$ if $K$ is an independent set in $D$ and for every vertex $v\in V\setminus K$, $v$ is at most distance 2 from $K$. In 1974, Chv\'atal and Lov\'asz proved that…

A quasi-kernel of a digraph $D$ is an independent set $Q$ such that every vertex can reach $Q$ in at most two steps. A 48-year conjecture made by P.L. Erd\H{o}s and Sz\'ekely, denoted the small QK conjecture, says that every sink-free…

A directed graph $D=(V(D),A(D))$ has a kernel if there exists an independent set $K\subseteq V(D)$ such that every vertex $v\in V(D)-K$ has an ingoing arc $u\mathbin{\longrightarrow}v$ for some $u\in K$. There are directed graphs that do…

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…

Given a directed graph G=(V,E) an independent set A of the vertices V is called quasi-kernel (quasi-sink) iff for each point v there is a path of length at most 2 from some point of A to v (from v to some point of A). Every finite directed…

Any directed graph $D=(V(D),A(D))$ in this work is assumed to be finite and without self-loops. A source in a directed graph is a vertex having at least one ingoing arc. A quasi-kernel $Q\subseteq V(D)$ is an independent set in $D$ such…

Given a digraph $D$, we say that a set of vertices $Q\subseteq V(D)$ is a $q$-kernel if $Q$ is an independent set and if every vertex of $D$ can be reached from $Q$ by a path of length at most $q$. In this paper, we initiate the study of…

In a digraph $D$,a quasi-kernel is an independent set $Q$ such that for every vertex $u$, there is a vertex $v \in Q$ satisfying $\text{dist}(v,u)\leq 2$. In 1974 Chv\'atal and Lov\'asz showed every digraph contains a quasi-kernel. In 1976,…

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…

Let $D$ be a digraph. We call a subset $N$ of $V(D)$ $k$-independent if for every pair of vertices $u,v \in N$, $d(u,v) \geq k$; and we call it $\ell$-absorbent if for every vertex $u \in V(D) \setminus N$, there exists $v \in N$ such that…

Let $D = (V(D), A(D))$ be a digraph. A subset $S \subseteq V(D)$ is $k$-independent if the distance between every pair of vertices of $S$ is at least $k$, and it is $\ell$-absorbent if for every vertex $u$ in $V(D) \setminus S$ there exists…

Let $k$ be an integer with $k\geq 2$. A $k$-king in a digraph $D$ is a vertex which can reach every other vertex by a directed path of length at most $k$ and a non-king is a vertex which is not a 3-king. A subset $K$ is $k$-independent if…

A kernel in a digraph is an independent and absorbent subset of its vertex set. A digraph is critical kernel imperfect if it does not have a kernel, but every proper induced subdigraph does. In this article, we characterize asymmetrical…

Quasi-cliques are dense incomplete subgraphs of a graph that generalize the notion of cliques. Enumerating quasi-cliques from a graph is a robust way to detect densely connected structures with applications to bio-informatics and social…

In a directed graph, a kernel is a subset of vertices that is both stable and absorbing. Not all digraphs have a kernel, but a theorem due to Boros and Gurvich guarantees the existence of a kernel in every clique-acyclic orientation of a…

A set of n non-collinear points in the Euclidean plane defines at least n different lines. Chen and Chv\'tal in 2008 conjectured that the same results is true in metric spaces for an adequate definition of line. More recently, it was…

A vertex of a graph is bisimplicial if the set of its neighbors is the union of two cliques; a graph is quasi-line if every vertex is bisimplicial. A recent result of Chudnovsky and Seymour asserts that every non-empty even-hole-free graph…

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…