Abstract
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∈Vf(v). The double Roman domination number γdR(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 γdR(P(5k,k))=8k for k≡2,3mod5 and 8k≤γdR(P(5k,k))≤8k+2 for k≡0,1,4mod5. We also improve the upper bounds for generalized Petersen graphs P(20k,k).
Highlights
Double Roman domination of graphs was first studied in [1], motivated by a number of applications of Roman domination in present time and in history [2]
We improve the upper bounds for generalized Petersen graphs P(20k, k)
The main result of our paper are either exact values or narrow bounds for the double Roman domination numbers of all Petersen graphs P(5k, k)
Summary
Double Roman domination of graphs was first studied in [1], motivated by a number of applications of Roman domination in present time and in history [2]. Linear time algorithms exist for interval graphs and block graphs [8], for trees [10], for proper interval graphs [11] and for unicyclic graphs [9] Another avenue of research that is motivated by high complexity of the problem is to obtain closed expressions for the double Roman domination number of some families of graphs. Defined a graph G to be double Roman if γdR(G) = 3γ(G), where γ(G) is the domination number of G. Roman domination and double Roman domination is a rather new variety of interest [1,2,7,24,25,26,27]
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