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All Journal Electronic Journal of Graph Theory and Applications (EJGTA)
Doost Ali Mojdeh
Department of Mathematics, University of Mazandaran, Babolsar, Iran
Electrical & Electronics Engineering

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On the distance domination number of bipartite graphs Doost Ali Mojdeh; Seyed Reza Musawi; Esmaeil Nazari
Electronic Journal of Graph Theory and Applications (EJGTA) Vol 8, No 2 (2020): Electronic Journal of Graph Theory and Applications
Publisher : GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.5614/ejgta.2020.8.2.11

Abstract

‎A subset D ⊆ V(G) is called a k-distance dominating set of G if every vertex in V(G)-D is within distance k from some vertex of D‎. ‎The minimum cardinality among all k-distance dominating sets of G is called the k-distance domination number of G. ‎In this note we give upper bounds on the k-distance domination number of a connected bipartite graph‎, ‎and improve some results have been given like Theorems 2.1 and 2.7 in [Tian and Xu‎, ‎A note on distance domination of graphs‎, ‎Australasian Journal of Combinatorics‎, ‎43 (2009)‎, ‎181-190]‎. 
Unique response strong Roman dominating functions of graphs Doost Ali Mojdeh; Guoliang Hao; Iman Masoumi; Ali Parsian
Electronic Journal of Graph Theory and Applications (EJGTA) Vol 9, No 2 (2021): Electronic Journal of Graph Theory and Applications
Publisher : GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.5614/ejgta.2021.9.2.18

Abstract

Given a simple graph G=(V,E) with maximum degree Δ. Let (V0, V1, V2) be an ordered partition of V, where Vi = {v ∈ V : f(v)=i} for i = 0, 1 and V2 = {v ∈ V : f(v)≥2}. A function f : V → {0, 1, …, ⌈Δ/2⌉+1} is a strong Roman dominating function (StRDF) on G, if every v ∈ V0 has a neighbor w ∈ V2 and f(w)≥1 + ⌈1/2|N(w)∩V0|⌉. A function f : V → {0, 1, …, ⌈Δ/2⌉+1} is a unique response strong Roman function (URStRF), if w ∈ V0, then |N(w)∩V2|≤1 and w ∈ V1 ∪ V2 implies that |N(w)∩V2|=0. A function f : V → {0, 1, …, ⌈Δ/2⌉+1} is a unique response strong Roman dominating function (URStRDF) if it is both URStRF and StRDF. The unique response strong Roman domination number of G, denoted by uStR(G), is the minimum weight of a unique response strong Roman dominating function. In this paper we approach the problem of a Roman domination-type defensive strategy under multiple simultaneous attacks and begin with the study of several mathematical properties of this invariant. We obtain several bounds on such a parameter and give some realizability results for it. Moreover, for any tree T of order n ≥ 3 we prove the sharp bound uStR(T)≤8n/9.
Outer independent global dominating set of trees and unicyclic graphs Doost Ali Mojdeh; Mortaza Alishahi
Electronic Journal of Graph Theory and Applications (EJGTA) Vol 7, No 1 (2019): Electronic Journal of Graph Theory and Applications
Publisher : GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.5614/ejgta.2019.7.1.10

Abstract

Let G be a graph. A set D ⊆ V(G) is a global dominating set of G if D is a dominating set of G and $\overline G$. γg(G) denotes global domination number of G. A set D ⊆ V(G) is an outer independent global dominating set (OIGDS) of G if D is a global dominating set of G and V(G) − D is an independent set of G. The cardinality of the smallest OIGDS of G, denoted by γgoi(G), is called the outer independent global domination number of G. An outer independent global dominating set of cardinality γgoi(G) is called a γgoi-set of G. In this paper we characterize trees T for which γgoi(T) = γ(T) and trees T for which γgoi(T) = γg(T) and trees T for which γgoi(T) = γoi(T) and the unicyclic graphs G for which γgoi(G) = γ(G), and the unicyclic graphs G for which γgoi(G) = γg(G).
Co-Authors Ali Parsian Esmaeil Nazari Guoliang Hao Iman Masoumi Mortaza Alishahi Seyed Reza Musawi