[1, 2]-Sets in Graphs

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A subset S⊆V in a graph G=(V,E) is a [j,k]-set if, for every vertex vεV\-S, j≤|N(v)\∩S|≤k for non-negative integers j and k, that is, every vertex vεV\-S is adjacent to at least j but not more than k vertices in S. In this paper, we focus on small j and k, and relate the concept of [j,k]-sets to a host of other concepts in domination theory, including perfect domination, efficient domination, nearly perfect sets, 2-packings, and k-dependent sets. We also determine bounds on the cardinality of minimum [1, 2]-sets, and investigate extremal graphs achieving these bounds. This study has implications for restrained domination as well. Using a result for [1, 3]-sets, we show that, for any grid graph G, the restrained domination number is equal to the domination number of G.