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All Journal Electronic Journal of Graph Theory and Applications (EJGTA)
Nader Jafari Rad
Department of Mathematics, Shahed University, Tehran, Iran
Electrical & Electronics Engineering

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On the complexity of some hop domination parameters Nader Jafari Rad; Elahe Shabani
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.2019.7.1.6

Abstract

A hop Roman dominating function (HRDF) on a graph G = (V, E) is a function f : V → {0, 1, 2} having the property that for every vertex v ∈ V with f(v) = 0 there is a vertex u with f(u) = 2 and d(u, v) = 2. The weight of an HRDF f is the sum of its values on V. The minimum weight of an HRDF on G is called the hop Roman domination number of G. An HRDF f is a hop Roman independent dominating function (HRIDF) if for any pair v, w with f(v) > 0 and f(w) > 0, d(v, w) ≠ 2. The minimum weight of an HRIDF on G is called the hop Roman independent domination number of G. In this paper, we study the complexity of the hop independent dominating problem, the hop Roman domination function problem and the hop Roman independent domination function problem, and show that the decision problem for each of the above three problems is NP-complete even when restricted to planar bipartite graphs or planar chordal graphs.
On the outer-independent double Italian domination number Noor A'lawiah Abd Aziz; Hailiza Kamarulhaili; Farzaneh Azvin; Nader Jafari Rad
Electronic Journal of Graph Theory and Applications (EJGTA) Vol 10, No 2 (2022): 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.2022.10.2.2

Abstract

‎An outer-independent Italian dominating function (OIIDF) on a graph G is a function f : V(G)→{0, 1, 2} such that every vertex v ∈ V(G) with f(v)=0 has at least two neighbors assigned 1 under f or one neighbor w with f(w)=2, and the set {u ∈ V(G)|f(u)=0} is independent. An outer-independent double Italian dominating function (OIDIDF) on a graph G is a function f : V(G)→{0, 1, 2, 3} such that if f(v)∈{0, 1} for a vertex v ∈ V(G), then ∑u ∈ N[v]f(u)≥3 and the set {u ∈ V(G)|f(u)=0} is independent. The weight of an OIIDF (respectively, OIDIDF) f is the value w(f)=∑v ∈ V(G)f(v). The minimum weight of an OIIDF (respectively, OIDIDF) on a graph G is called the outer-independent Italian (respectively, outer-independent double Italian) domination number of G. We characterize all trees T with outer-independent double Italian domination number twice the outer-independent Italian domination number. We also present lower bounds on the outer-independent double Italian domination number of a connected graph G in terms of the order, minimum and maximum degrees.
Co-Authors Elahe Shabani Farzaneh Azvin Hailiza Kamarulhaili Noor A'lawiah Abd Aziz