Independent Rainbow Domination Numbers of Generalized Petersen Graphs P ( n,2) and P ( n,3)

Boštjan Gabrovšek, Aljoša Peperko and Janez Žerovnik
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Boštjan Gabrovšek: FME, University of Ljubljana, Aškerčeva 6, SI-1000 Ljubljana, Slovenia
Aljoša Peperko: FME, University of Ljubljana, Aškerčeva 6, SI-1000 Ljubljana, Slovenia
Janez Žerovnik: FME, University of Ljubljana, Aškerčeva 6, SI-1000 Ljubljana, Slovenia

Mathematics, 2020, vol. 8, issue 6, 1-13

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.

Keywords: independent rainbow domination; independent rainbow domination number; generalized Petersen graphs; tropical algebra; path algebra (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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