The probabilistic approach to limited packings in graphs

Abstract

We consider (closed neighbourhood) packings and their generalization in graphs.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 probabilistic approach 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 ∆,.The problem of finding a maximum size k-limited packing is known to be N Pcomplete even in split or bipartite graphs.

(Open) packing number of some graph products

“…Let P G be a ρ(G − )-set, let P H be a ρ(H)-set, and let I be the set of all singletons of G. For a vertex v ∈ V (H), we set P = (P G ×{v})∪(I ×P H ). According to (8), and the paragraph after it, we can deduce that for any two vertices (g, h), (g ′ , h ′ ) ∈ P , it follows that d G•H ((g, h), (g ′ , h ′ )) ≥ 3.…”

“…A generalization of packings presented in [9] is called k-limited packing, where the closed neighborhood of every vertex can have at most k vertices in a k-limited packing set S. They exhibited some real-world applications of it to network security, market saturation, NIMBY and codes. A probabilistic approach to some bounds of the k-limited packings can be found in [8]. More results on this topic can be found in [23].…”

Incidence dimension and 2-packing number in graphs

“…A generalization of packing, called k-limited packing, is presented in [8]. Here every vertex can have at most k neighbors in a k-limited packing set S. To achieve some bounds, a probabilistic approach to k-limited packings was introduced in [7]. A further generalization of it is shown in [3].…”

Graphs with unique maximum packing of closed neighborhoods

“…A generalization of packing presented in [6] is the k-limited packing where every vertex can have at most k neighbors in a k-limited packing set S. A probabilistic approach to k-limited packings can be found in [5]. A further generalization, that is, the generalized limited packing of the k-limited packing, see [3], brings a dynamic approach with respect to the vertices of G, where v ∈ V (G) can have a different number of neighbors k v for every vertex v in a generalized limited packing.…”