Browsing by Author "Mojdeh, Doost Ali"

Now showing 1 - 5 of 5

Access status: Open Access ,
k-perfect geodominating sets in graphs
(2007) Mojdeh, Doost Ali; Rad, Nader Jafari
A perfect geodominating set in a graph $G$ is a geodominating set $S$ such that any vertex $v \in V(G)\setminus S$ is geodominated by exactly one pair of vertices of $S$. A $k$-perfect geodominating set is a geodominating set $S$ such that any vertex $v \in V(G)\setminus S$ is geodominated by exactly one pair $x$, $y$ of vertices of $S$ with $d(x, y) = k$. We study perfect and $k$-perfect geodomination numbers of a graph $G$.
Access status: Open Access ,
On the diameter of dot-critical graphs
(2009) Mojdeh, Doost Ali; Mirzamani, Somayeh
A graph $G$ is $k$-dot-critical (totaly $k$-dot-critical) if $G$ is dot-critical (totaly dot-critical) and the domination number is $k$. In the paper [T. Burtona, D. P. Sumner, <i>Domination dot-critical graphs</i>, Discrete Math, 306(2006), 11–18] the following question is posed: What are the best bounds for the diameter of a $k$-dot-critical graph and a totally $k$-dot-critical graph $G$ with no critical vertices for $k \geq 4$? We find the best bound for the diameter of a $k$-dot-critical graph, where $k \in\{4,5,6\}$ and we give a family of $k$-dot-critical graphs (with no critical vertices) with sharp diameter $2k - 3$ for even $k \geq 4$.
Access status: Open Access ,
On the inverse signed total domination number in graphs
(2017) Mojdeh, Doost Ali; Samadi, Babak
In this paper, we study the inverse signed total domination number in graphs and present new sharp lower and upper bounds on this parameter. For example by making use of the classic theorem of Turán (1941), we present a sharp upper bound on $K_{r+1}$-free graphs for $r\geq 2$. Also, we bound this parameter for a tree from below in terms of its order and the number of leaves and characterize all trees attaining this bound.
Access status: Open Access ,
Outer independent rainbow dominating functions in graphs
(Wydawnictwa AGH, 2020) Mansouri, Zhila; Mojdeh, Doost Ali
A 2-rainbow dominating function (2-rD function) of a graph $G=(V,E)$ is a function $f:V(G)\rightarrow\{\emptyset,\{1\},\{2\},\{1,2\}\}$ having the property that if $f(x)=\emptyset$, then $f(N(x))=\{1,2\}$. The 2-rainbow domination number $\gamma_{r2}(G)$ is the minimum weight of $\sum_{v\in V(G)}|f(v)|$ taken over all 2-rainbow dominating functions $f$. An outer-independent 2-rainbow dominating function (OI2-rD function) of a graph $G$ is a 2-rD function $f$ for which the set of all $v \in V(G)$ with $f(v)=\emptyset$ is independent. The outer independent 2-rainbow domination number $\gamma_{oir2}(G)$ is the minimum weight of an OI2-rD function of $G$. In this paper, we study the OI2-rD number of graphs. We give the complexity of the problem OI2-rD of graphs and present lower and upper bounds on $\gamma_{oir2}(G)$. Moreover, we characterize graphs with some small or large OI2-rD numbers and we also bound this parameter from above for trees in terms of the order, leaves and the number of support vertices and characterize all trees attaining the bound. Finally, we show that any ordered pair $(a,b)$ is realizable as the vertex cover number and OI2-rD numbers of some non-trivial tree if and only if $a+1\leq b\leq 2a$.
Access status: Open Access ,
Strong geodomination in graphs
(2008) Rad, Nader Jafari; Mojdeh, Doost Ali
A pair $x$, $y$ of vertices in a nontrivial connected graph $G$ is said to geodominate a vertex $v$ of $G$ if either $v \in \{x, y\}$ or $v$ lies in an $x - y$ geodesic of $G$. A set $S$ of vertices of $G$ is a geodominating set if every vertex of $G$ is geodominated by some pair of vertices of $S$. In this paper we study strong geodomination in a graph $G$.