Abstract
We obtain new results on 3-rainbow domination numbers of generalized Petersen graphs P(6k,k). In some cases, for some infinite families, exact values are established; in all other cases, the lower and upper bounds with small gaps are given. We also define singleton rainbow domination, where the sets assigned have a cardinality of, at most, one, and provide analogous results for this special case of rainbow domination.
Highlights
We study a special case where vertices are colored by sets with, at most, one color
In cases when n = 6k, on the other hand, we have the following upper bound for generalized Petersen graphs [26]
An interesting special case are tRD functions that assign only singletons or empty sets. We call such functions singleton tRD functions (StRD functions) and the minimal weight obtained when considering only StRD functions singleton t-rainbow domination number denoted by γrt
Summary
3-Rainbow Domination Number of Generalized Petersen Graphs P(6k, k ). Inspired by several facility location problems, Brešar, Henning and Rall [1,2,3] initiated the study of the k-rainbow domination problem. The problem is proved to be NP-complete, even if the input graph is a chordal graph or a bipartite graph [2]. This variation of the general domination problem has already attracted considerable attention. The growing interest in domination problems [4] is based on a variety of practical applications on one hand and, on the other hand, expected (and usually proven) intractability on general graphs. In [5], 2-rainbow domination of generalized Petersen graphs was extensively studied. We recall the definitions and some other preliminary material from [5]
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