Related papers: Orientable domination in product-like graphs

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…

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…

The secure domination numbers of the Cartesian products of two small graphs with paths or cycles is determined, as well as for Mobius ladder graphs. Prior to this work, in all cases where the secure domination number has been determined,…

In this article we give a new definition of direct product of two arbitrary fuzzy graphs. We define the concepts of domination and total domination in this new product graph. We obtain an upper bound for the total domination number of the…

Let $G$ be a graph with vertex set $V(G)$. A function $f:V(G)\rightarrow \{0,1,2\}$ is a Roman dominating function on $G$ if every vertex $v\in V(G)$ for which $f(v)=0$ is adjacent to at least one vertex $u\in V(G)$ such that $f(u)=2$. The…

Let $G$ be a simple graph of order $n$. The domination polynomial of $G$ is the polynomial $D(G,\lambda)=\sum_{i=0}^{n} d(G,i) \lambda^{i}$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$. Every root of $D(G,\lambda)$ is…

Ho proved in [A note on the total domination number, Util.Math. 77 (2008) 97--100] that the total domination number of the Cartesian product of any two graphs with no isolated vertices is at least one half of the product of their total…

A longest sequence $(v_1,\ldots,v_k)$ of vertices of a graph $G$ is a Grundy total dominating sequence of $G$ if for all $i$, $N(v_i) \setminus \bigcup_{j=1}^{i-1}N(v_j)\not=\emptyset$. The length $k$ of the sequence is called the Grundy…

A \emph{domatic} (\emph{total domatic}) \emph{$k$-coloring} of a graph $G$ is an assignment of $k$ colors to the vertices of $G$ such that each vertex contains vertices of all $k$ colors in its closed neighborhood (neighborhood). The…

The dominating number $\gamma(G)$ of a graph $G$ is the minimum size of a vertex set whose closed neighborhoods cover all vertices of $G$, while the packing number $\rho(G)$ is the maximum size of a vertex set whose closed neighborhoods are…

Given a graph G, the domination number gamma(G) of G is the minimum order of a set S of vertices such that each vertex not in S is adjacent to some vertex in S. Equivalently, label the vertices from {0, 1} so that the sum over each closed…

A graph is well-dominated if all its minimal dominating sets have the same cardinality. In this paper we prove that at least one factor of every connected, well-dominated Cartesian product is a complete graph, which then allows us to give a…

Power domination is a two-step observation process that is used to monitor power networks and can be viewed as a combination of domination and zero forcing. Given a graph $G$, a subset $S\subseteq V(G)$ that can observe all vertices of $G$…

The open neighbourhood of a vertex $v$ of a graph $G$ is the set $N(v)$ consisting 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…

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.…

Given a graph~$G$, the domination number, denoted by~$\gamma(G)$, is the minimum cardinality of a dominating set in~$G$. Dual to the notion of domination number is the packing number of a graph. A packing of~$G$ is a set of vertices whose…

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 $\gamma(G)$…

A graph is said to be well-edge-dominated if all its minimal edge dominating sets are minimum. It is known that every well-edge-dominated graph $G$ is also equimatchable, meaning that every maximal matching in $G$ is maximum. In this paper,…

The unitary Cayley graph of $\mathbb{Z}/n\mathbb{Z}$, denoted $X_n$, is the graph on $\{0,\dots,n-1\}$ where vertices $a$ and $b$ are adjacent if and only if $\gcd(a-b,n) = 1$. We answer a question of Defant and Iyer by constructing a…

A set $S\subseteq V$ is a dominating set of $G$ if every vertex in $V - S$ is adjacent to at least one vertex in $S$. The domination number $\gamma(G)$ of $G$ equals the minimum cardinality of a dominating set $S$ in $G$; we say that such a…