相关论文: On the Concentration of the Domination Number of t…

This paper delves into the stability of the $2$-domination number in simple undirected graphs. The $2$-domination number of a graph $G$, $\gamma_2(G)$, represents the minimum size of a vertex subset where every other vertex in the graph is…

The chromatic number of a very dense random graph $G(n,p)$, with $p \ge 1 - n^{-c}$ for some constant $c > 0$, was first studied by Surya and Warnke, who conjectured that the typical deviation of $\chi(G(n,p))$ from its mean is of order…

A directed dominating set in a directed graph $D$ is a set $S$ of vertices of $V$ such that every vertex $u \in V(D) \setminus S$ has an adjacent vertex $v$ in $S$ with $v$ directed to $u$. The directed domination number of $D$, denoted by…

Denote by $\mathcal{H}_k (n,p)$ the random $k$-graph in which each $k$-subset of $\{1... n\}$ is present with probability $p$, independent of other choices. More or less answering a question of Balogh, Bohman and Mubayi, we show: there is a…

Majority dynamics on the binomial Erd\H{o}s-R\'enyi graph $\mathsf{G}(n,p)$ with $p=\lambda/\sqrt{n}$ is studied. In this process, each vertex has a state in $\{0,1\}$ and at each round, every vertex adopts the state of the majority of its…

We study majority dynamics on the binomial random graph $G(n,p)$ with $p = d/n$ and $d > \lambda n^{1/2}$, for some large $\lambda>0$. In this process, each vertex has a state in $\{-1,+1 \}$ and at each round every vertex adopts the state…

Let $G=(V,E)$ be a simple connected graph. A set of vertices $S\subseteq V$ is said to be a dominating set if for any vertex in $V\setminus S$ is adjacent to at least one vertex in $S$. The domination number $\gamma(G)$ of $G$ is the…

Let $G = (V, E)$ be a simple graph of order $n$. The total dominating set of $G$ is a subset $D$ of $V$ that every vertex of $V$ is adjacent to some vertices of $D$. The total domination number of $G$ is equal to minimum cardinality of…

The theory of dense graph limits comes with a natural sampling process which yields an inhomogeneous variant G(n,W) of the Erdos-Renyi random graph. Here we study the clique number of these random graphs. We establish the concentration of…

The domatic number of a graph $G$, denoted $dom(G)$, is the maximum possible cardinality of a family of disjoint sets of vertices of $G$, each set being a dominating set of $G$. It is well known that every graph without isolated vertices…

This papers focuses on the average order of dominating sets of a graph. We find the extremal graphs for the maximum and minimum value over all graphs on $n$ vertices, while for trees we prove that the star minimizes the average order of…

For graphs $G$ and $F$, the saturation number $\textit{sat}(G,F)$ is the minimum number of edges in an inclusion-maximal $F$-free subgraph of $G$. In 2017, Kor\'andi and Sudakov initiated the study of saturation in random graphs. They…

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 locating-dominating set in an undirected graph is a subset of vertices $S$ such that $S$ is dominating and for every $u,v \notin S$, we have $N(u)\cap S\ne N(v)\cap S$. In this paper, we consider the oriented version of the problem. A…

We commence the study of domination in the incidence graphs of combinatorial designs. Let $D$ be a combinatorial design and denote by $\gamma(D)$ the domination number of the incidence (Levy) graph of $D$. We obtain a number of results…

The k-domination number of a graph is the minimum size of a set X such that every vertex of G is in distance at most k from X. We give a linear time constant-factor approximation algorithm for k-domination number in classes of graphs with…

A directed dominating set in a directed graph $D$ is a set $S$ of vertices of $V$ such that every vertex $u \in V(D) \setminus S$ has an adjacent vertex $v$ in $S$ with $v$ directed to $u$. The directed domination number of $D$, denoted by…

For any graph $G$ of order $n$ with degree sequence $d_{1}\geq\cdots\geq d_{n}$, we define the double Slater number $s\ell_{\times2}(G)$ as the smallest integer $t$ such that $t+d_{1}+\cdots+d_{t-e}\geq2n-p$ in which $e$ and $p$ are the…

The study of domination in graphs has led to a variety of domination problems studied in the literature. Most of these follow the following general framework: Given a graph $G$ and an integer $k$, decide if there is a set $S$ of $k$…

In a graph $G$, a vertex dominates itself and its neighbors. A subset $S\subseteq V(G)$ is said to be a double dominating set of $G$ if $S$ dominates every vertex of $G$ at least twice. The double domination number $\gamma_{\times 2}(G)$ is…