Strong Equality of Perfect Roman and Weak Roman Domination in Trees

Abdollah Alhevaz, Mahsa Darkooti, Hadi Rahbani and Yilun Shang
Additional contact information
Abdollah Alhevaz: Faculty of Mathematical Sciences, Shahrood University of Technology, P.O. Box 316-3619995161, Shahrood, Iran
Mahsa Darkooti: Faculty of Mathematical Sciences, Shahrood University of Technology, P.O. Box 316-3619995161, Shahrood, Iran
Hadi Rahbani: Faculty of Mathematical Sciences, Shahrood University of Technology, P.O. Box 316-3619995161, Shahrood, Iran
Yilun Shang: Department of Computer and Information Sciences, Northumbria University, Newcastle NE1 8ST, UK

Mathematics, 2019, vol. 7, issue 10, 1-13

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 ∈ V f ( v ) . The weak (resp., perfect) Roman domination number, denoted by γ r ( G ) (resp., γ R p ( 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.

Keywords: Perfect Roman dominating function; Roman dominating number; weak Roman dominating function (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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