Nader Jafari Rad
Shahrood University of Technology
[ 1 ] - Total $k$-Rainbow domination numbers in graphs
Let $k\geq 1$ be an integer, and let $G$ be a graph. A $k$-rainbow dominating function (or a {$k$-RDF) of $G$ is a function $f$ from the vertex set $V(G)$ to the family of all subsets of $\{1,2,\ldots ,k\}$ such that for every $v\in V(G)$ with $f(v)=\emptyset $, the condition $\bigcup_{u\in N_{G}(v)}f(u)=\{1,2,\ldots,k\}$ is fulfilled, where $N_{G}(v)$ is the open neighborhood of $v$. The weight …
[ 2 ] - On Hop Roman Domination in Trees
Let $G=(V,E)$ be a graph. A subset $S\subset V$ is a hop dominating set if every vertex outside $S$ is at distance two from a vertex of $S$. A hop dominating set $S$ which induces a connected subgraph is called a connected hop dominating set of $G$. The connected hop domination number of $G$, $ \gamma_{ch}(G)$, is the minimum cardinality of a connected hop dominating set of $G$. A hop Roman domin…
[ 3 ] - Bounds on the outer-independent double Italian domination number
An outer-independent double Italian dominating function (OIDIDF) on a graph $G$ with vertex set $V(G)$ is a function $f:V(G)\longrightarrow \{0,1,2,3\}$ such that if $f(v)\in\{0,1\}$ for a vertex $v\in V(G)$ then $\sum_{u\in N[v]}f(u)\geq3$, and the set $ \{u\in V(G)|f(u)=0\}$ is independent. The weight of an OIDIDF $f$ is the value $w(f)=\sum_{v\in V(G)}f(v)$. The minimum weight of an OIDIDF on …
[ 4 ] - Total domination in cubic Knödel graphs
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 $G$. A set $D\subseteq V(G)$ is a total dominating set if for each $u\in V(G)$, $u$ is adjacent to some vertex $v\in D$. The total domination number, $\gamma_{t}(G)$ of…
[ 5 ] - Strength of strongest dominating sets in fuzzy graphs
A set S of vertices in a graph G=(V,E) is a dominating set of G if every vertex of V-S is adjacent to some vertex of S. For an integer k≥1, a set S of vertices is a k-step dominating set if any vertex of $G$ is at distance k from some vertex of S. In this paper, using membership values of vertices and edges in fuzzy graphs, we introduce the concepts of strength of strongest dominating set as well…