Document Type
Article
Publication Date
6-1-2019
Description
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.
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This work is licensed under a Creative Commons Attribution-Share Alike 4.0 International License.
Citation Information
Chellali, Mustapha; Haynes, Teresa W.; and Hedetniemi, Stephen T.. 2019. k-Efficient Partitions of Graphs. Communications in Combinatorics and Optimization. Vol.4(2). 109-122. https://doi.org/10.22049/CCO.2019.26446.1112 ISSN: 2538-2128
Copyright Statement
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.