Related papers: A Note on Connected Dominating Set in Graphs Witho…

The power domination problem focuses on finding the optimal placement of phase measurement units (PMUs) to monitor an electrical power network. In the context of graphs, the power domination number of a graph $G$, denoted $\gamma_P(G)$, is…

Motivated by Thomassen's well-known line graph conjecture, many researchers have explored sufficient conditions for claw-free graphs to be Hamiltonian or Hamilton-connected. In 1994, Ageev proved that every $2$-connected claw-free graph…

We propose the conjecture that the domination number $\gamma(G)$ of a $\Delta$-regular graph $G$ with $\Delta\geq 1$ is always at most its edge domination number $\gamma_e(G)$, which coincides with the domination number of its line graph.…

Let $G=(V, E)$ be a graph. A set $S\subseteq V(G)$ is a {\it dominating set} of $G$ if every vertex in $V\setminus S$ is adjacent to a vertex of $S$. The {\it domination number} of $G$, denoted by $\gamma(G)$, is the cardinality of a…

The dominating number $\gamma(G)$ of a graph $G$ is the minimum size of a vertex set whose closed neighborhood covers all the vertices of the graph. The packing number $\rho(G)$ of $G$ is the maximum size of a vertex set whose closed…

While a number of bounds are known on the zero forcing number $Z(G)$ of a graph $G$ expressed in terms of the order of a graph and maximum or minimum degree, we present two bounds that are related to the (upper) total domination number…

Let $\gamma'_s(G)$ be the signed edge domination number of G. In 2006, Xu conjectured that: for any $2$-connected graph G of order $ n (n \geq 2),$ $\gamma'_s(G)\geq 1$. In this article we show that this conjecture is not true. More…

The following inequality chain $$ ir(G)\le \gamma(G)\le i(G)\le \alpha(G) \le \Gamma(G) \le I\!R(G)$$ is known as a domination chain, where $ir(G), \gamma(G), i(G), \alpha(G), \Gamma(G)$, and $I\!R(G)$ are the lower irredundance number, the…

A vertex set $S$ of a graph $G$ is a \emph{dominating set} if each vertex of $G$ either belongs to $S$ or is adjacent to a vertex in $S$. The \emph{domination number} $\gamma(G)$ of $G$ is the minimum cardinality of $S$ as $S$ varies over…

A dominating set in a graph $G$ is a set $S$ of vertices such that every vertex in $V(G) \setminus S$ is adjacent to a vertex in $S$. A restrained dominating set of $G$ is a dominating set $S$ with the additional restraint that the graph $G…

Let $G$ be a connected graph of order $n$ with vertex set $V(G)$. A subset $S\subseteq V(G)$ is an $(a,b)$-dominating set if every vertex $v\in S$ is adjacent to at least $a$ vertices in $S$ and every $v\in V\setminus S$ is adjacent to at…

A paired dominating set $P$ is a dominating set with the additional property that $P$ has a perfect matching. While the maximum cardainality of a minimal dominating set in a graph $G$ is called the upper domination number of $G$, denoted by…

For a graph $G$, a subset $S$ of $V(G)$ is a {\it hop dominating set} of $G$ if every vertex not in $S$ has a $2$-step neighbor in $S$. The {\it hop domination number}, $\gamma_h(G)$, of $G$ is the minimum cardinality of a hop dominating…

A set $S$ of vertices in a graph $G(V,E)$ is called a dominating set if every vertex $v\in V$ is either an element of $S$ or is adjacent to an element of $S$. A set $S$ of vertices in a graph $G(V,E)$ is called a total dominating set if…

The aim of this work is to investigate the nonnegative signed domination number $\gamma^{NN}_s$ with emphasis on regular, ($r+1$)-clique-free graphs and trees. We give lower and upper bounds on $\gamma^{NN}_s$ for regular graphs and prove…

A set $D$ of vertices of a graph $G$ is a dominating set of $G$ if every vertex in $V_G-D$ is adjacent to at least one vertex in $D$. The domination number (upper domination number, respectively) of a graph $G$, denoted by $\gamma(G)$…

Let $\gamma(G)$ be the domination number of a graph $G$. A graph $G$ is \emph{domination-vertex-critical}, or \emph{$\gamma$-vertex-critical}, if $\gamma(G-v)< \gamma(G)$ for every vertex $v \in V(G)$. In this paper, we show that: Let $G$…

Let $G$ be a graph with no isolated vertex. In this paper, we study a parameter that is a relaxation of arguably the most important domination parameter, namely the total domination number, $\gamma_t(G)$. A set $S$ of vertices in $G$ is a…

Let $G$ be a graph of order $n$. A classical upper bound for the domination number of a graph $G$ having no isolated vertices is $\lfloor\frac{n}{2}\rfloor$. However, for several families of graphs, we have $\gamma(G) \le…

Let $G=(V,E)$ be a connected, finite undirected graph. A set $S \subseteq V$ is said to be a total dominating set of $G$ if every vertex in $V$ is adjacent to some vertex in $S$. The total domination number, $\gamma_{t}(G)$, is the minimum…