Double Roman Domination in Generalized Petersen Graphs

The double Roman domination can be described as a strengthened defense strategy. In an empire, each city can be protected by at most three troops. Every city having no troops must be adjacent to at least two cities with two troops or one city with three troops. Every city having one troop must be adjacent to at least one city with more than one troop. Such an assignment is called a double Roman dominating function (DRDF) of an empire/a graph. The minimum number of troops under such an assignment is the double Roman domination number, denoted as γdR. Shao et al. (2018) determine the exact value of γdR(P(n,1)). Jiang et al. (2018) determine γdR(P(n,2)). In this article, we investigate the double Roman domination number of P(n,k) for k ≥ 3. We determine the exact value of γdR(P(n,k)) for n ≡ 0(mod4) and k ≡ 1(mod2), and present an improved upper bound of γdR(P(n,k)) for n ̸≡ 0(mod4) or k ̸≡ 1(mod2). Our results imply P(n,3) for n ≡ 0(mod4) is double Roman which can partially answer the open question present by Beeler et al. (2016).