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k-Efficient Partitions of Graphs

A set S = {u1, u2,..., ut} of vertices of G is an efficient dominating set if every vertex of G is dominated exactly once by the vertices of S. Letting Ui denote the set of vertices dominated by ui, we note that {U1, U2,... Ut} is a partition of the vertex set of G and that each Ui contains the vertex ui and all the vertices at distance 1 from it in G. In this paper, we generalize the concept of efficient domination by considering k-efficient domination partitions of the vertex set of G, where each element of the partition is a set consisting of a vertex ui and all the vertices at distance di from it, where di ∈ {0, 1,..., k}. For any integer k ≥ 0, the k-efficient domination number of G equals the minimum order of a k-efficient partition of G. We determine bounds on the k-efficient domination number for general graphs, and for k ∈ {1, 2}, we give exact values for some graph families. Complexity results are also obtained.

Identiferoai:union.ndltd.org:ETSU/oai:dc.etsu.edu:etsu-works-11232
Date01 June 2019
CreatorsChellali, Mustapha, Haynes, Teresa W., Hedetniemi, Stephen T.
PublisherDigital Commons @ East Tennessee State University
Source SetsEast Tennessee State University
Detected LanguageEnglish
Typetext
Formatapplication/pdf
SourceETSU Faculty Works
Rightshttp://creativecommons.org/licenses/by-sa/4.0/

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