The Double Roman Domination Numbers of Generalized Petersen Graphs P(n, 2)

Abstract

A double Roman dominating function (DRDF) f on a given graph G is a mapping from V ( G ) to { 0 , 1 , 2 , 3 } in such a way that a vertex u for which f ( u ) = 0 has at least a neighbor labeled 3 or two neighbors both labeled 2 and a vertex u for which f ( u ) = 1 has at least a neighbor labeled 2 or 3. The weight of a DRDF f is the value w ( f ) = ∑ u ∈ V ( G ) f ( u ) . The minimum weight of a DRDF on a graph G is called the double Roman domination number γ d R ( G ) of G. In this paper, we determine the exact value of the double Roman domination number of the generalized Petersen graphs P ( n , 2 ) by using a discharging approach.

Double Roman Graphs in P(3k, k)

“…Petersen graphs are among the most interesting examples when considering nontrivial graph invariants. The domination and its variations of generalized Petersen graphs have attracted considerable attention, see for example [21][22][23][24][25][26][27][28]. Let n and k be integers where n ≥ 3, k ≥ 1, and k < n 2 .…”

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.…”

Radio Number for Generalized Petersen Graphs $P(n,2)$

“…Peterson graphs has been largely studied in past years [29]- [31], for example, spectrum of generalized Petersen graphs has been studied in [31]. Coloring and Tutte polynomial of Peterson graphs have been given in [32] and [33] respectively.…”