Related papers: Domination number in the annihilating-submodule gr…

Let $G$ be a graph. A total dominating set of $G$ is a set $S$ of vertices of $G$ such that every vertex is adjacent to at least one vertex in $S$. The total domatic number of a graph is the maximum number of total dominating sets which…

In this paper, we associate a new topology to a nonzero unital module $M$ over a commutative $R$, which is called Golomb topology of the $R$-module $M$. Let $M\ $be an\ $R$-module and $B_{M}$ be the family of coprime cosets $\{m+N\}$ where…

Given a duo module $M$ over an associative (not necessarily commutative) ring $R,$ a Zariski topology is defined on the spectrum $\mathrm{Spec}^{\mathrm{fp}}(M)$ of {\it fully prime} $R$-submodules of $M$. We investigate, in particular, the…

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…

Let $G=(V,E)$ be a graph. A subset $D$ of $V(G)$ is called a super dominating set if for every $v \in V(G)-D$ there exists an external private neighbour of $v$ with respect to $V(G)-D.$ The minimum cardinality of a super dominating set is…

Let $G = (V, E)$ be a simple graph of order $n$. The total dominating set 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 total…

A set S of vertices in a graph G is a dominating set of G if every vertex not in S is adjacent to a vertex in S . The domination number of G, denoted by $\gamma$(G), is the minimum cardinality of a dominating set in G. In a breakthrough…

An integer distance digraph is the Cayley graph $\Gamma(\mathbb{Z},S)$ of the additive group $\mathbb{Z}$ of all integers with respect to a finite subset $S\subseteq\mathbb{Z}$. The domination ratio of $\Gamma(\mathbb{Z},S)$, defined as the…

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…

Given a graph $G$, a dominating set of $G$ is a set $S$ of vertices such that each vertex not in $S$ has a neighbor in $S$. The domination number of $G$, denoted $\gamma(G)$, is the minimum size of a dominating set of $G$. The independent…

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 $\gamma(G)$…

For a graph $G = (V,E),$ a subset $S$ of $V$ is a perfect dominating set of $G$ if every vertex not in $S$ is adjacent to exactly one vertex in $S.$ The perfect domination number, $\gamma_p(G),$ is the minimum cardinality of a perfect…

Given a $K$-vector space $V$, let $\sigma(V,K)$ denote the covering number, i.e. the smallest (cardinal) number of proper subspaces whose union covers $V$. Analogously, define $\sigma(M,R)$ for a module $M$ over a unital commutative ring…

A set $D$ of vertices in $G$ is a disjunctive dominating set in $G$ if every vertex not in $D$ is adjacent to a vertex of $D$ or has at least two vertices in $D$ at distance $2$ from it in $G$. The disjunctive domination number,…

Let $G$ be a finite simple group. By a theorem of Guralnick and Kantor, $G$ contains a conjugacy class $C$ such that for each non-identity element $x \in G$, there exists $y \in C$ with $G = \langle x,y\rangle$. Building on this deep…

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 dominating set $S$ in a graph $G$ is said to be perfect if every vertex of $G$ not in $S$ is adjacent to just one vertex of $S$. Given a vertex subset $S'$ of a side $P_m$ of an $m\times n$ grid graph $G$, the perfect dominating sets $S$…

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…

The comaximal graph $ \Gamma(R) $ of a commutative ring $R$ is a simple graph with vertex set $ R $ and two distinct vertices $ a $ and $b $ of $ \Gamma(R) $ are adjacent if and only if $ aR+bR=R $, where $ aR $ is the ideal generated by $…

The middle graph $M(G)$ of a graph $G$ is the graph obtained by subdividing each edge of $G$ exactly once and joining all these newly introduced vertices of adjacent edges of $G$. A perfect Roman dominating function on a graph $G$ is a…