Restrained double Roman domination of a graph
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by
Doost Ali Mojdeh, Iman Masoumi, Lutz Volkmann
2021
Abstract
For a graph G=(V,E), a restrained double Roman dominating function is a
function f:V→{0,1,2,3} having the property that if f(v)=0, then the
vertex v must have at least two neighbors assigned 2 under f or one neighbor w
with f(w)=3, and if f(v)=1, then the vertex v must have at least one neighbor w
with f(w)≥2, and at the same time, the subgraph G[V_0] which includes
vertices with zero labels has no isolated vertex. The weight of a restrained
double Roman dominating function f is the sum f(V)=∑_v∈Vf(v), and the
minimum weight of a restrained double Roman dominating function on G is the
restrained double Roman domination number of G. We initiate the study of
restrained double Roman domination with proving that the problem of computing
this parameter is NP-hard. Then we present an upper bound on the restrained
double Roman domination number of a connected graph G in terms of the order of
G and characterize the graphs attaining this bound. We study the restrained
double Roman domination versus the restrained Roman domination. Finally, we
characterized all trees T attaining the exhibited bound.
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