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- W2896652362 abstract "Abstract In a finite undirected graph G, a vertex v dominates itself and its neighbors. A vertex set D is an efficient dominating set (e.d.s. for short) of G if every vertex of G is dominated by exactly one vertex of D. The Efficient Domination (ED) problem, which asks for the existence of an e.d.s. in G, is well known to be NP -complete for graphs of maximum degree 3. We say that an e.d.s. D of G is k-bounded (k-b.e.d.s. for short) if the degree of every vertex in D is at most k in G. The task of the k-Bounded Weighted Efficient Domination (k-BWED) problem is to determine whether a given vertex-weighted graph G admits a k-b.e.d.s., and if so, to compute one of minimum weight. It easily follows from the NP -completeness of ED for graphs of maximum degree 3 that the k-BWED problem is NP -complete for every k ≥ 3 , and clearly, the k-BWED problem is solvable in linear time for k ≤ 1 . In this note, we show that the 2-BWED problem is solvable in time O ( | V ( G ) | ( | V ( G ) | + | E ( G ) | ) ) , thus obtaining a dichotomy of the complexity status of k-BWED over all k ≥ 0 ." @default.
- W2896652362 created "2018-10-26" @default.
- W2896652362 creator A5027970574 @default.
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- W2896652362 date "2019-02-01" @default.
- W2896652362 modified "2023-10-15" @default.
- W2896652362 title "A dichotomy for weighted efficient dominating sets with bounded degree vertices" @default.
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- W2896652362 doi "https://doi.org/10.1016/j.ipl.2018.10.007" @default.
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