Related papers: On the maximum number of minimum dominating sets i…

A set $S$ of vertices in $G$ is a semitotal dominating set of $G$ if it is a dominating set of $G$ and every vertex in $S$ is within distance $2$ of another vertex of $S$. The \emph{semitotal domination number}, $\gamma_{t2}(G)$, is the…

The $3/5$-conjecture for the domination game states that the game domination numbers of an isolate-free graph $G$ on $n$ vertices are bounded as follows: $\gamma_g(G)\leq \frac{3n}5 $ and $\gamma_g'(G)\leq \frac{3n+2}5 $. Recent progress…

In this paper, we investigate the relation between the (fractional) domination number of a digraph $G$ and the independence number of its underlying graph, denoted by $\alpha(G)$. More precisely, we prove that every digraph $G$ has…

The minimum status of a graph is the minimum of statuses of all vertices of this graph. We give a sharp upper bound for the minimum status of a connected graph with fixed order and matching number (domination number, respectively), and…

Let $ G $ be a graph. A subset $S \subseteq V(G) $ is called a total dominating set if every vertex of $G$ is adjacent to at least one vertex of $S$. The total domination number, $\gamma_{t}$($G$), is the minimum cardinality of a total…

It is conjectured that the game domination number is at most $3n/5$ for every $n$-vertex graph which does not contain isolated vertices. It was proved in the recent years that the conjecture holds for several graph classes, including the…

A polynomial is said to be unimodal if its coefficients are non-decreasing and then non-increasing. The domination polynomial of a graph $G$ is the generating function of the number of dominating sets of each cardinality in $G$. In…

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…

Fix a positive integer $n$ and consider the bipartite graph whose vertices are the $3$-element subsets and the $2$-element subsets of $[n]=\{1,2,\dots,n\}$, and there is an edge between $A$ and $B$ if $A\subset B$. We prove that the…

We propose the following conjecture: For every fixed $\alpha\in [0,\frac 13)$, each graph of minimum degree at least $(1+\alpha)\frac k2$ and maximum degree at least $2(1-\alpha)k$ contains each tree with $k$ edges as a subgraph. Our main…

A subset $S\subseteq V$ in a graph $G=(V,E)$ is a total $[1,2]$-set if, for every vertex $v\in V$, $1\leq |N(v)\cap S|\leq 2$. The minimum cardinality of a total $[1,2]$-set of $G$ is called the total $[1,2]$-domination number, denoted by…

We analyze the domination game, where two players, Dominator and Staller, construct together a dominating set M in a given graph, by alternately selecting vertices into M. Each move must increase the size of the dominated set. The players…

We prove that for every graph $G$ on $n$ vertices and with minimum degree five, the domination number $\gamma(G)$ cannot exceed $n/3$. The proof combines an algorithmic approach and the discharging method. Using the same technique, we…

As a natural variant of domination in graphs, Dankelmann et al. [Domination with exponential decay, Discrete Math. 309 (2009) 5877-5883] introduce exponential domination, where vertices are considered to have some dominating power that…

Let $\gamma(G)$ and $\gamma_t(G)$ denote the domination number and the total domination number, respectively, of a graph $G$ with no isolated vertices. It is well-known that $\gamma_t(G) \leq 2\gamma(G)$. We provide a characterization of a…

We prove that the class of trees with unique minimum edge-vertex dominating sets is equivalent to the class of trees with unique minimum paired dominating sets.

A locating-dominating set (LDS) of a graph $G$ is a dominating set $S$ of $G$ such that for every two vertices $u$ and $v$ in $V(G) \setminus S$, $N(u)\cap S \neq N(v)\cap S$. The locating-domination number $\gamma^{L}(G)$ is the minimum…

We notice that Haynes-Hedetniemi-Slater Conjecture is true (i.e. $\gamma(G) \leq \frac{\delta}{3\delta -1}n$ for every graph $G$ of size $n$ with minimum degree $\delta \geq 4$, where $\gamma(G)$ is the domination number of $G$). Because…

Domination of grids has been proved to be a demanding task and with the addition of independence it becomes more challenging. It is known that no grid with $m,n \geq 5$ has an efficient dominating set, also called perfect code, that is, an…

Let $G=(V,E)$ be a graph. A subset $D$ of $V$ is a \textit{restrained dominating set} if every vertex in $V \setminus D$ is adjacent to a vertex in $D$ and to a vertex in $V \setminus D$. The \textit{restrained domination number}, denoted…