Related papers: Safe sets in digraphs

Suppose one needs to change the direction of at least $\epsilon n^2$ edges of an $n$-vertex tournament $T$, in order to make it $H$-free. A standard application of the regularity method shows that in this case $T$ contains at least…

A homogeneous set of an $n$-vertex graph is a set $X$ of vertices ($2\le |X|\le n-1$) such that every vertex not in $X$ is either complete or anticomplete to $X$. A graph is called prime if it has no homogeneous set. A chain of length $t$…

Let $G=(V,E)$ be a simple graph. A dominating set of $G$ is a subset $S\subseteq V$ such that every vertex not in $S$ is adjacent to at least one vertex in $S$. The cardinality of a smallest dominating set of $G$, denoted by $\gamma(G)$, is…

A graph $G$ is perfectly divisible if every induced subgraph $H$ of $G$ contains a set $X$ of vertices such that $X$ meets all largest cliques of $H$, and $X$ induces a perfect graph. The chromatic number of a perfectly divisible graph $G$…

Given a digraph $D$, we denote by $\vec{\alpha}(D)$ the maximum size of an acyclic set of $D$ (i.e. a set of vertices which induces a subdigraph with no directed cycles), and by $\vec\chi(D)$ the minimum number of acyclic sets into which…

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…

Let $G$ be a graph. A set $S$ of vertices in $G$ dominates the graph if every vertex of $G$ is either in $S$ or a neighbor of a vertex in $S$. Finding a minimal cardinality set which dominates the graph is an NP-complete problem. The graph…

A dominating set of a graph $G=(V,E)$ is a subset of vertices $S\subseteq V$ such that every vertex $v\in V\setminus S$ has at least one neighbor in set $S$. The corresponding optimization problem is known to be NP-hard. The best known…

A graph $G$ is called well-covered if all maximal independent sets of vertices have the same cardinality. A simplicial complex $\Delta$ is called pure if all of its facets have the same cardinality. Let $\mathcal G$ be the class of graphs…

Let $A$ and $B$ be disjoint, non-adjacent vertex-sets in an undirected, connected graph $G$, whose vertices are associated with positive weights. We address the problem of identifying a minimum-weight subset of vertices $S\subseteq V(G)$…

A graph $G$ is said to be a `set graph' if it admits an acyclic orientation that is also `extensional', in the sense that the out-neighborhoods of its vertices are pairwise distinct. Equivalently, a set graph is the underlying graph of the…

For a given undirected graph $G$, an \emph{ordered} subset $S = {s_1,s_2,...,s_k} \subseteq V$ of vertices is a resolving set for the graph if the vertices of the graph are distinguishable by their vector of distances to the vertices in…

A {\em kernel by properly colored paths} of an arc-colored digraph $D$ is a set $S$ of vertices of $D$ such that (i) no two vertices of $S$ are connected by a properly colored directed path in $D$, and (ii) every vertex outside $S$ can…

In the Maximum Independent Set problem we are asked to find a set of pairwise nonadjacent vertices in a given graph with the maximum possible cardinality. In general graphs, this classical problem is known to be NP-hard and hard to…

Given a simple undirected graph $G = (V, E)$, the open neighbourhood of a vertex $v \in V$ is defined as $N_G(v) = \{u \in V \mid uv \in E\}$, and the closed neighbourhood as $N_G[v] = N_G(v) \cup \{v\}$. A subset $D \subseteq V$ is called…

The neighbourhood of a vertex $v$ of a graph $G$ is the set $N(v)$ of all vertices adjacent to $v$ in $G$. For $D\subseteq V(G)$ we define $\overline{D}=V(G)\setminus D$. A set $D\subseteq V(G)$ is called a super dominating set if for every…

A hole is a chordless cycle with at least four vertices. A hole is odd if it has an odd number of vertices. A dart is a graph which vertices $a, b, c, d, e$ and edges $ab, bc, bd, be, cd, de$. Dart-free graphs have been actively studied in…

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…

A set $D \subseteq V$ for the graph $G=(V, E)$ is called a dominating set if any vertex $v\in V\setminus D$ has at least one neighbor in $D$. Fomin et al.[9] gave an algorithm for enumerating all minimal dominating sets with $n$ vertices in…

A vertex set $D$ in a finite undirected graph $G$ is an {\em efficient dominating set} (\emph{e.d.s.}\ for short) of $G$ if every vertex of $G$ is dominated by exactly one vertex of $D$. The \emph{Efficient Domination} (ED) problem, which…