Abstract
We obtain new results on independent 2- and 3-rainbow domination numbers of generalized Petersen graphs P ( n , k ) for certain values of n , k ∈ N . By suitably adjusting and applying a well established technique of tropical algebra (path algebra) we obtain exact 2-independent rainbow domination numbers of generalized Petersen graphs P ( n , 2 ) and P ( n , 3 ) thus confirming a conjecture proposed by Shao et al. In addition, we compute exact 3-independent rainbow domination numbers of generalized Petersen graphs P ( n , 2 ) . The method used here is developed for rainbow domination and for Petersen graphs. However, with some natural modifications, the method used can be applied to other domination type invariants, and to many other classes of graphs including grids and tori.
Highlights
As a combinatorial optimization problem, ordinary domination consists of determining the minimum number of places in which to keep a resource such that every place either has a resource or is adjacent to the place in which the resource exists
In this article we have studied the t-independent rainbow domination number irt of polygraphs
For a polygraph that is formed by n monographs we have established that its t-independent rainbow domination number equals the minimum weight of a closed walk of length n of a suitably associated graph
Summary
As a combinatorial optimization problem, ordinary domination consists of determining the minimum number of places in which to keep a resource such that every place either has a resource or is adjacent to the place in which the resource exists. In this article we obtain new results on exact values of independent 2-rainbow domination number of generalized Petersen graphs P(n, 2) and P(n, 3). Theorems 6 and 7), where it was conjectured that the upper bounds (obtained in [2] (Theorems 6 and 7)) for the independent 2-rainbow domination number of generalized Petersen graphs P(n, 2). We confirm this by suitably adjusting and applying a well known tropical (path) algebra technique for polygraphs (see e.g., [3,4,5,6]). By applying this technique we obtain the exact formula for the independent 3-rainbow domination number of generalized. In the concluding section we discuss the potential of the method to be generalized to other domination type invariants and to other classes of graphs
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