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 u for which f ( u ) = 0 has at least a neighbor labeled 3 or two neighbors both labeled 2 and a vertex u for which f ( u ) = 1 has at least a neighbor labeled 2 or 3. The weight of a DRDF f is the value w ( f ) = ∑ u ∈ V ( G ) f ( u ) . The minimum weight of a DRDF on a graph G is called the double Roman domination number γ d R ( G ) of G. In this paper, we determine the exact value of the double Roman domination number of the generalized Petersen graphs P ( n , 2 ) by using a discharging approach.
Highlights
In this paper, only graphs without multiple edges or loops are considered
For a vertex subset S ⊆ V ( G ), we denote by G [S] the subgraph induced by S
For a set S = { x1, x2, · · ·, xn }, if xi = x j for some i and j, S is considered as a multiset
Summary
Only graphs without multiple edges or loops are considered. For two vertices u and v of a graph G, we say u ∼ v in G if uv ∈ E( G ). A double Roman dominating function (DRDF) f on a given graph G is a mapping from V ( G ) In a double Roman dominating function of weight γdR ( G ), no vertex needs to be assigned the value one.
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