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

A double Roman dominating function on a graph G=(V,E) is a function f:V & RARR;{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)= n-ary sumation( v & ISIN;V)f(v). The double Roman domination number gamma dR(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(5k,k). It is proven that gamma(dR)(P(5k,k))=8k for k & EQUIV;2,3mod5 and 8k & LE;gamma(dR)(P(5k,k))& LE;8k+2 for k & EQUIV;0,1,4mod5. We also improve the upper bounds for generalized Petersen graphs P(20k,k).