Related papers: Domination, matching and transversal numbers for B…

The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex labeling (edge labeling) with $d$ labels that is preserved only by a trivial automorphism. A set $S$ of vertices in $G$…

A domination coloring of a graph $G$ is a proper vertex coloring of $G$ such that each vertex of $G$ dominates at least one color class, and each color class is dominated by at least one vertex. The minimum number of colors among all…

A subset $D$ of vertices of a graph $G$ is a dominating set if for each $u\in V(G)\setminus D$, $u$ is adjacent to some vertex $v\in D$. The domination number, $\gamma(G)$ of $G$, is the minimum cardinality of a dominating set of $G$. For…

A semitotal dominating set of a graph $G$ with no isolated vertex is a dominating set $D$ of $G$ such that every vertex in $D$ is within distance two of another vertex in $D$. The minimum size $\gamma_{t2}(G)$ of a semitotal dominating set…

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…

A set $S$ of vertices in a graph $G$ is a dominating set if every vertex not in $S$ is adjacent to a vertex in $S$. If, in addition, $S$ is an independent set, then $S$ is an independent dominating set. The independent domination number…

Understanding graph density profiles is notoriously challenging. Even for pairs of graphs, complete characterizations are known only in very limited cases, such as edges versus cliques. This paper explores a relaxation of the graph density…

Given a graph $G$ such that each vertex $v_i$ has a value $f(v_i)$, the expanded-clique graph $H$ is the graph where each vertex $v_i$ of $G$ becomes a clique $V_i$ of size $f(v_i)$ and for each edge $v_iv_j \in E(G)$, there is a vertex of…

Graph energy and Domination in graphs are most studied areas of graph theory. In this paper we made an attempt to connect these two areas of graph theory by introducing c-dominating energy of a graph $G$. First, we show the chemical…

For a graph $G,$ the set $D \subseteq V(G)$ is a porous exponential dominating set if $1 \le \sum_{d \in D} \left( 2 \right)^{1-dist(d,v)}$ for every $v \in V(G),$ where $dist(d,v)$ denotes the length of the shortest $dv$ path. The porous…

For a given graph $G=(V,E)$ and permutation $\pi:V\mapsto V$ the prism $\pi G$ of $G$ is defined as follows: $V(\pi G)=V(G)\cup V(G')$, where $G'$ is a copy of $G$, and $E(\pi G)=E(G)\cup E(G')\cup M_{\pi}$, where $M_{\pi}=\{uv': u\in V(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…

Given a graph $G$, a subset $S$ of vertices of $G$ is an efficient dominating set ($EDS$) if $|N[v] \cap S|=1,$ for all $v\in V(G)$. A graph $G$ is efficiently dominatable if it possesses an $EDS$. The efficient domination number of G is…

In this paper, we prove a problem proposed by Bre\v{s}ar: for any graphs $G$ and $H$, $\Gamma(G\square H)\ge\Gamma(G)\Gamma(H)+ \min\{|V(G)|-\Gamma(G),|V(H)|-\Gamma(H)\}$, where $\Gamma(G)$ denotes the upper domination number of $G$.

A dominating set of a graph $G$ is a set $D \subseteq V(G)$ such that every vertex in $V(G) \setminus D$ is adjacent to at least one vertex in $D$. A set $L\subseteq V(G)$ is a locating set of $G$ if every vertex in $V(G) \setminus L$ has…

Let $G$ be a graph each component of which has order at least 3, and let $G$ have order $n$, size $m$, total domination number $\gamma_t$ and maximum degree $\Delta(G)$. Let $\Delta = 3$ if $\Delta(G) = 2$ and $\Delta = \Delta (G)$ if…

A dominating set of a graph $G$ is a set $S \subseteq V(G)$ such that every vertex in $V(G) \setminus S$ has a neighbor in $S$, where two vertices are neighbors if they are adjacent. A secure dominating set of $G$ is a dominating set $S$ of…

For $k \geq 1$ and a graph $G$ without isolated vertices, a \emph{total (distance) $k$-dominating set} of $G$ is a set of vertices $S \subseteq V(G)$ such that every vertex in $G$ is within distance $k$ to some vertex of $S$ other than…

The total domination number $\gamma_{t}(G)$ of a graph $G$ is the cardinality of a smallest set $D\subseteq V(G)$ such that each vertex of $G$ has a neighbor in $D$. The annihilation number $a(G)$ of $G$ is the largest integer $k$ such that…

Let $G$ be a simple graph of order $n$. The domination polynomial of $G$ is the polynomial $D(G, x)=\sum_{i=1}^n d(G,i) x^i$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$. For two graphs $G$ and $H$, let $\mathcal{C} =…