Perfect and quasiperfect domination in trees
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A k quasip erfect dominating set of a connected graph G is a vertex subset S such that every vertex not in S is adjacent to at least one and at most k vertices in S . The cardinality of a minimum k-quasip erfect dominating set in G is denoted by 1 k ( G ) . These graph parameters were rst intro duced by Chellali et al. (2013) as a generalization of b oth the p erfect domination numb er 11 ( G ) and the domination numb er ( G ) . The study of the so-called quasip erfect domination chain 11 ( G ) 12 ( G ) 1 ( G ) = ( G ) enable us to analyze how far minimum dominating sets are from b eing p erfect. In this pap er, we provide, for any tree T and any p ositive integer k , a tight upp er b ound of 1 k ( T ) . We also prove that there are trees satisfying all p ossible equalities and inequalities in this chain. Finally a linear algorithm for computing 1 k ( T ) in any tree T is presente
CitationCáceres, José, Hernando, M., Mora, M., Pelayo, I. M., Luz Puertas, M. Perfect and quasiperfect domination in trees. "Applicable analysis and discrete mathematics", 25 Abril 2016, vol. 10, p. 46-64.