arXiv subjects

Doost Ali Mojdeh

Publications and source records attributed to Doost Ali Mojdeh.

27 records · Page 2Linked to original sources

arXiv2019

Double domination and total $2$-domination in digraphs and their dual problems

A subset $S$ of vertices of a digraph $D$ is a double dominating set (total $2$-dominating set) if every vertex not in $S$ is adjacent from at least two vertices in $S$, and every vertex in $S$ is adjacent from at least one vertex in $S$ (the subdigraph induced by $S$ has no isolated vertices). The double domination number (total $2$-domination number) of a digraph $D$ is the minimum cardinality of a double dominating set (total $2$-dominating set) in $D$. In this work, we investigate these concepts which can be considered as two extensions of double domination in graphs to digraphs, along with the concepts $2$-limited packing and total $2$-limited packing which have close relationships with the above-mentioned concepts.

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arXiv2019

Global offensive $k$-alliances in digraphs

In this paper, we initiate the study of global offensive $k$-alliances in digraphs. Given a digraph $D=(V(D),A(D))$, a global offensive $k$-alliance in a digraph $D$ is a subset $S\subseteq V(D)$ such that every vertex outside of $S$ has at least one in-neighbor from $S$ and also at least $k$ more in-neighbors from $S$ than from outside of $S$, by assuming $k$ is an integer lying between two minus the maximum in-degree of $D$ and the maximum in-degree of $D$. The global offensive $k$-alliance number $γ_{k}^{o}(D)$ is the minimum cardinality among all global offensive $k$-alliances in $D$. In this article we begin the study of the global offensive $k$-alliance number of digraphs. For instance, we prove that finding the global offensive $k$-alliance number of digraphs $D$ is an NP-hard problem for any value $k\in \{2-Δ^-(D),\dots,Δ^-(D)\}$ and that it remains NP-complete even when restricted to bipartite digraphs when we consider the non-negative values of $k$ given in the interval above. Based on these facts, lower bounds on $γ_{k}^{o}(D)$ with characterizations of all digraphs attaining the bounds are given in this work. We also bound this parameter for bipartite digraphs from above. For the particular case $k=1$, an immediate result from the definition shows that $γ(D)\leq γ_{1}^{o}(D)$ for all digraphs $D$, in which $γ(D)$ stands for the domination number of $D$. We show that these two digraph parameters are the same for some infinite families of digraphs like rooted trees and contrafunctional digraphs. Moreover, we show that the difference between $γ_{1}^{o}(D)$ and $γ(D)$ can be arbitrary large for directed trees and connected functional digraphs.

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arXiv2019

Independent double Roman domination in graphs

An independent double Roman dominating function (IDRDF) on a graph $G=(V,E)$ is a function $f:V(G)\rightarrow \{0,1,2,3\}$ having the property that if $f(v)=0$, then the vertex $v$ has at least two neighbors assigned $2$ under $f$ or one neighbor $w$ with assigned $3$ under $f$, and if $f(v)=1$, then there exists $w\in N(v)$ with $f(w)\geq2$ such that the positive weight vertices are independent. The weight of an IDRDF is the value $\sum_{u\in V}f(u)$. The independent double Roman domination number $i_{dR}(G)$ of a graph $G$ is the minimum weight of an IDRDF on G. We initiate the study of the independent double Roman domination and show its relationships to both independent domination number (IDN) and independent Roman $\{2\}$-domination number (IR2DN). We present several sharp bounds on the IDRDN of a graph $G$ in terms of the order of $G$, maximum degree and the minimum size of edge cover. Finally, we show that, any ordered pair $(a,b)$ is realizable as the IDN and IDRDN of some non-trivial tree if and only if $2a + 1 \le b \le 3a$.

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arXiv2018

On three outer-independent domination related parameters in graphs

Let $G$ be a graph and let $S\subseteq V(G)$. The set $S$ is a double outer-independent dominating set of $G$ if $|N[v]\cap D|\geq2$, for all $v\in V(G)$, and $V(G)\setminus S$ is independent. Similarly, $S$ is a $2$-outer-independent dominating set, if every vertex from $V(G)\setminus S$ has at least two neighbors in $S$ and $V(G)\setminus S$ is independent. Finally, $S$ is a total outer-independent dominating set if every vertex from $V(G)$ has a neighbor in $S$ and the complement of $S$ is an independent set. The double, total or $2$-outer-independent domination number of $G$ is the smallest possible cardinality of any double, total or $2$-outer-independent dominating set of $G$, respectively. In this paper, the $2$-outer-independent, the total outer-independent and the double outer-independent domination numbers of graphs are investigated. We prove some Nordhaus-Gaddum type inequalities, derive their computational complexity and present several bounds for them.

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arXiv2018

Bounds on the nonnegative signed domination number of graphs

The aim of this work is to investigate the nonnegative signed domination number $γ^{NN}_s$ with emphasis on regular, ($r+1$)-clique-free graphs and trees. We give lower and upper bounds on $γ^{NN}_s$ for regular graphs and prove that $n/3$ is the best possible upper bound on this parameter for a cubic graph of order $n$, specifically. As an application of the classic theorem of Turán we bound $γ^{NN}_s(G)$ from below, for an ($r+1$)-clique-free graph $G$ and characterize all such graphs for which the equality holds, which corrects and generalizes a result for bipartite graphs in [Electron. J. Graph Theory Appl. 4 (2) (2016), 231--237], simultaneously. Also, we bound $γ^{NN}_s(T)$ for a tree $T$ from above and below and characterize all trees attaining the bounds.

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arXiv2018

Some notes on the signed bad number in bipartite graphs

In this paper, we deal with the signed bad number and the negative decision number of graphs. We show that two upper bounds concerning these two parameters for bipartite graphs in papers [Discrete Math. Algorithms Appl. 1 (2011), 33--41] and [Australas. J. Combin. 41 (2008), 263--272] are not true as they stand. We correct them by presenting more general bounds for triangle-free graphs by using the classic theorem of Mantel from the extremal graph theory and characterize all triangle-free graphs attaining these bounds.

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arXiv2018

Packing and domination parameters in digraphs

Given a digraph $D=(V,A)$, a set $B\subset V$ is a packing set in $D$ if there are no arcs joining vertices of $B$ and for any two vertices $x,y\in B$ the sets of in-neighbors of $x$ and $y$ are disjoint. The set $S$ is a dominating set (an open dominating set) in $D$ if every vertex not in $S$ (in $V$) has an in-neighbor in $S$. Moreover, a dominating set $S$ is called a total dominating set if the subgraph induced by $S$ has no isolated vertices. The packing sets of maximum cardinality and the (total, open) dominating sets of minimum cardinality in digraphs are studied in this article. We prove that the two optimal sets concerning packing and domination achieve the same value for directed trees, and give some applications of it. We also show analogous equalities for all connected contrafunctional digraphs, and characterize all such digraphs $D$ for which such equalities are satisfied. Moreover, sharp bounds on the maximum and the minimum cardinalities of packing and dominating sets, respectively, are given for digraphs. Finally, we present solutions for two open problems, concerning total and open dominating sets of minimum cardinality, pointed out in [Australas. J. Combin. 39 (2007), 283--292].

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arXiv2018

Total domination in cubic Knödel graphs

A subset $D$ of vertices of a graph $G$ is a \textit{dominating set} if for each $u\in V(G)\setminus D$, $u$ is adjacent to some vertex $v\in D$. The \textit{dominating number}, $γ(G)$ of $G$, is the minimum cardinality of a dominating set of $G$. A set $D\subseteq V(G)$ is a \textit{total dominating set} if for each $u\in V(G)$, $u$ is adjacent to some vertex $v\in D$. the The \textit{total dominating number}, $γ_t(G)$ of $G$, is the minimum cardinality of a total dominating set of $G$. For an even integer $n\ge2$ and $1\leΔ\le\lfloor\log_2n\rfloor$, a \textit{Knödel graph} $W_{Δ,n}$ is a $Δ$-regular bipartite graph of even order $n$, with vertices $(i,j)$, for $i=1,2$ and $0\le j\le n/2-1$, where for every $j$,$0\le j\le n/2-1$,there is an edge between vertex $(1,j)$ and every vertex $(2,j+2^k-1 \text{(mod(n/2)})$, for $k=0,1,\cdots,Δ-1$. In this paper, we determine the total domination number in $3$-regular Knödel graphs $W_{3,n}$.

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arXiv2018

Domination in 4-regular 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, $γ(G)$ of $G$, is the minimum cardinality of a dominating set of $G$. For an even integer $n\ge2$ and $1\leΔ\le\lfloor\log_2n\rfloor$, a Knödel graph $W_{Δ,n}$ is a $Delta$-regular bipartite graph of even order $n$, with vertices$(i,j)$, for $i=1,2$ and $0\le j\le n/2-1$, where for every $j$, $0\le j\le n/2-1$, there is an edge between $(1,j)$ and $(2,j+2^k-1 \text{(mod(n/2)})$, for $k=0,1,\cdots,Δ-1$. In this paper, we determine the domination number in $4$-regular Knödel graphs $W_{4,n}$.

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