On the Double Roman Domination in Spider Graphs

Abstract

A double Roman dominating function (DRDF) f on a given graph G is a mapping from V ( G ) to {0, 1, 2, 3} in such a way that a vertex v for which f ( v ) = 0 has at least a neighbor labeled 3 or two neighbors both labeled 2 and a vertex v for which f ( v ) = 1 has at least a neighbor labeled 2 or 3. The weight of a DRDF f is the value w ( f ) = ∑ v ∈ V ( G ) f ( v ) . The minimum weight of a DRDF on a graph G is called the double Roman domination number of G . In this paper, we determine the exact value of the double Roman domination number of the Spider graphs S m , 2 and S m , 3 , and obtain an upper bound of the Spider graphs S m , n .