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The probabilistic approach to limited packings in graphs

Zverovich, Vadim; Gagarin, Andrei

The probabilistic approach to limited packings in graphs Thumbnail


Andrei Gagarin


© 2014 Elsevier B.V. All rights reserved. We consider (closed neighbourhood) packings and their generalization in graphs. A vertex set X in a graph G is a k-limited packing if for every vertex vεV(G), |N[v]∩X|≤k, where N[v] is the closed neighbourhood of v. The k-limited packing number (G) of a graph G is the largest size of a k-limited packing in G. Limited packing problems can be considered as secure facility location problems in networks. In this paper, we develop a new application of the probabilistic method to limited packings in graphs, resulting in lower bounds for the k-limited packing number and a randomized algorithm to find k-limited packings satisfying the bounds. In particular, we prove that for any graph G of order n with maximum vertex degree δ,(G)≥kn(k+1)(δk)(δ+1)k. Also, some other upper and lower bounds for (G) are given.


Zverovich, V., & Gagarin, A. (2015). The probabilistic approach to limited packings in graphs. Discrete Applied Mathematics, 184, 146-153.

Journal Article Type Article
Publication Date Jan 1, 2015
Publicly Available Date Jun 6, 2019
Journal Discrete Applied Mathematics
Print ISSN 0166-218X
Publisher Elsevier
Peer Reviewed Peer Reviewed
Volume 184
Pages 146-153
Keywords k-limited packings, the probabilistic method, lower and upper bounds, randomized algorithm
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