TY - JOUR

T1 - Strong Equality of Perfect Roman and Weak Roman Domination in Trees

AU - Alhevaz, Abdollah

AU - Darkooti, Mahsa

AU - Rahbani, Hadi

AU - Shang, Yilun

PY - 2019/10/21

Y1 - 2019/10/21

N2 - 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.

AB - 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.

KW - Perfect Roman dominating function

KW - Roman dominating number

KW - weak Roman dominating function

U2 - 10.3390/math7100997

DO - 10.3390/math7100997

M3 - Article

SN - 2227-7390

VL - 7

JO - Mathematics

JF - Mathematics

IS - 10

M1 - 997

ER -