On the Double Roman Domination in Generalized Petersen Graphs P (5 k , k )

A double Roman dominating function on a graph G = ( V , E ) is a function f : V → { 0 , 1 , 2 , 3 } satisfying the condition that every vertex u for which f ( u ) = 0 is adjacent to at least one vertex assigned 3 or at least two vertices assigned 2, and every vertex u with f ( u ) = 1 is adjacent to at least one vertex assigned 2 or 3. The weight of f equals w ( f ) = ∑ v ∈ V f ( v ) . The double Roman domination number γ d R ( G ) of a graph G equals the minimum weight of a double Roman dominating function of G . We obtain closed expressions for the double Roman domination number of generalized Petersen graphs P ( 5 k , k ) . It is proven that γ d R ( P ( 5 k , k ) ) = 8 k for k ≡ 2 , 3 mod 5 and 8 k ≤ γ d R ( P ( 5 k , k ) ) ≤ 8 k + 2 for k ≡ 0 , 1 , 4 mod 5 . We also improve the upper bounds for generalized Petersen graphs P ( 20 k , k ) .