Related papers: Extremal digraphs for open neighbourhood location-…

A dominating set of a graph $G=(V,E)$ is a vertex set $D$ such that every vertex in $V(G) \setminus D$ is adjacent to a vertex in $D$. The cardinality of a smallest dominating set of $D$ is called the domination number of $G$ and is denoted…

Let $G$ be a graph with vertex set $V$, and let $k$ be a positive integer. A set $D \subseteq V$ is a \emph{distance-$k$ dominating set} of $G$ if, for each vertex $u \in V-D$, there exists a vertex $w\in D$ such that $d(u,w) \le k$, where…

A proper vertex-colouring of a graph G is said to be locally identifying if for any pair u,v of adjacent vertices with distinct closed neighbourhoods, the sets of colours in the closed neighbourhoods of u and v are different. We show that…

Introduced by Albertson et al. \cite{albertson}, the distinguishing number $D(G)$ of a graph $G$ is the least integer $r$ such that there is a $r$-labeling of the vertices of $G$ that is not preserved by any nontrivial automorphism of $G$.…

A subset $S$ of vertices of a digraph $D$ is a double dominating set (total $2$-dominating set) if every vertex not in $S$ is adjacent from at least two vertices in $S$, and every vertex in $S$ is adjacent from at least one vertex in $S$…

A non-empty subset $S$ of the vertices of a digraph $D$ is called a {\it safe set} if \begin{itemize} \item[(i)] for every strongly connected component $M$ of $D-S$, there exists a strongly connected component $N$ of $D[S]$ such that there…

We propose to study a problem that arises naturally from both Topological Numbering of Directed Acyclic Graphs, and Additive Coloring (also known as Lucky Labeling). Let $D$ be a digraph and $f$ a labeling of its vertices with positive…

Let $G$ be a graph of a network system with vertices, $V(G)$, representing physical locations and edges, $E(G)$, representing informational connectivity. A \emph{locating-dominating (LD)} set $S \subseteq V(G)$ is a subset of vertices…

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…

A locating-dominating set of a graph $G$ is a dominating set of $G$ such that every vertex of $G$ outside the dominating set is uniquely identified by its neighborhood within the dominating set. The location-domination number of $G$ is the…

A graph is called a sum graph if its vertices can be labelled by distinct positive integers such that there is an edge between two vertices if and only if the sum of their labels is the label of another vertex of the graph. Most papers on…

The concept of majority domination in graphs has been defined in at least two different ways: As a function and as a set. In this work we extend the latter concept to digraphs, while the former was extended in another paper. Given a digraph…

A set $D\subseteq V$ of a graph $G=(V,E)$ is called a neighborhood total dominating set of $G$ if $D$ is a dominating set and the subgraph of $G$ induced by the open neighborhood of $D$ has no isolated vertex. Given a graph $G$,…

Given a digraph $G$, a set $X\subseteq V(G)$ is said to be absorbing set (resp. dominating set) if every vertex in the graph is either in $X$ or is an in-neighbour (resp. out-neighbour) of a vertex in $X$. A set $S\subseteq V(G)$ is said to…

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…

Local Irregularity Conjecture states that every simple connected graph, except special cacti, can be decomposed into at most three locally irregular graphs, i.e., graphs in which adjacent vertices have different degrees. The connected…

Let $G=(V,E)$ be a graph. A subset $D\subseteq V$ is a dominating set if every vertex not in $D$ is adjacent to a vertex in $D$. A dominating set $D$ is called a total dominating set if every vertex in $D$ is adjacent to a vertex in $D$.…

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…

Let $G=(V,E)$ be a graph. A set $S\subseteq V(G)$ is a dominating set, if every vertex in $V(G)\backslash S$ is adjacent to at least one vertex in $S$. The $k$-dominating graph of $G$, $D_k (G)$, is defined to be the graph whose vertices…

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…