Strong Equality of Perfect Roman and Weak Roman Domination in Trees

Alhevaz, Abdollah, Darkooti, Mahsa, Rahbani, Hadi and Shang, Yilun (2019) Strong Equality of Perfect Roman and Weak Roman Domination in Trees. Mathematics, 7 (10). p. 997. ISSN 2227-7390

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Official URL: http://dx.doi.org/10.3390/math7100997

Abstract

Let G=(V,E) be a graph and f:V⟶{0,1,2} be a function. Given a vertex u with f(u)=0, if all neighbors of u have zero weights, then u is called undefended with respect to f. Furthermore, if every vertex u with f(u)=0 has a neighbor v with f(v)>0 and the function f′:V⟶{0,1,2} with f′(u)=1, f′(v)=f(v)−1, f′(w)=f(w) if w∈V∖{u,v} has no undefended vertex, then f is called a weak Roman dominating function. Also, the function f is a perfect Roman dominating function if every vertex u with f(u)=0 is adjacent to exactly one vertex v for which f(v)=2. Let the weight of f be w(f)=∑v∈Vf(v). The weak (resp., perfect) Roman domination number, denoted by γr(G) (resp., γpR(G)), is the minimum weight of the weak (resp., perfect) Roman dominating function in G. In this paper, we characterize those trees where the perfect Roman domination number strongly equals the weak Roman domination number, in the sense that each weak Roman dominating function of minimum weight is, at the same time, perfect Roman dominating.

Item Type: Article
Uncontrolled Keywords: Perfect Roman dominating function; Roman dominating number; weak Roman dominating function
Subjects: G100 Mathematics
Department: Faculties > Engineering and Environment > Computer and Information Sciences
Depositing User: Paul Burns
Date Deposited: 21 Oct 2019 09:26
Last Modified: 21 Oct 2019 09:30
URI: http://nrl.northumbria.ac.uk/id/eprint/41173

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