Perfect Roman domination in trees

Perfect double Roman domination of trees

“…A perfect Roman dominating f unction of a graph G is defined in [12] as a function Let F be the family of all trees T whose vertex set can be partitioned into sets, each set inducing a P 5 on five vertices, such that the subgraph induced by the central vertices of these P 5 's is connected. We call the subtree induced by these central vertices the underlying subtree of the resulting tree T , and we call each such path P 5 a base path of the tree T .…”

The Italian Domination Numbers of Generalized Petersen Graphs P(n,3)

“…Since then, a lot of papers have been published on various aspects of Roman domination [3,4] and many variations [5][6][7]. Italian domination is a new variation of Roman domination.…”

“…As an interesting family of graphs, various types of domination on generalized Petersen graphs have been studied extensively [15][16][17][18]. For two natural numbers n and k with n ≥ 3 and 1 ≤ k ≤ n−1 2 , the generalized Petersen graph P(n, k) is a graph on 2n vertices, V = {v i |0 ≤ i ≤ 2n − 1} and E = {v i v i+1 , v i v i+2 |0 ≤ i ≤ 2n − 1 and i = 0 (mod 2)} ∪ {v i v i+2k |0 ≤ i ≤ 2n − 1 and i = 1 (mod 2)}, where subscripts are taken modulo n. Figure 1a shows the graph of P(n, 3); for convenience and clarity, we always cut the graph P(n, 3) between v 0 and v 2n−2 , and Figure 1b shows the cut P (7,3). demonstrate that the Italian domination number is equal to the 2-rainbow domination number for trees and cactus graphs with no even cycles.…”

Strong Equality of Perfect Roman and Weak Roman Domination in Trees

“…Henning, Klostermeyer and MacGillivray [9] introduced the concept of perfect Roman domination in graphs. An RDF f = (V 0 , V 1 , V 2 ) is referred to as a perfect Roman dominating function (PRDF) when each vertex u with f (u) = 0 has only one neighbor v such that f (v) = 2 [9]. The minimum weight of an RDF is represented by γ P R (G) and is called the perfect Roman domination number.…”