17 research outputs found
The probabilistic approach to limited packings in graphs
© 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
Limited packings of closed neighbourhoods in graphs
The k-limited packing number, , of a graph , introduced by
Gallant, Gunther, Hartnell, and Rall, is the maximum cardinality of a set
of vertices of such that every vertex of has at most elements of
in its closed neighbourhood. The main aim in this paper is to prove the
best-possible result that if is a cubic graph, then , improving the previous lower bound given by Gallant, \emph{et al.}
In addition, we construct an infinite family of graphs to show that lower
bounds given by Gagarin and Zverovich are asymptotically best-possible, up to a
constant factor, when is fixed and tends to infinity. For
tending to infinity and tending to infinity sufficiently
quickly, we give an asymptotically best-possible lower bound for ,
improving previous bounds
New facets of the 2-dominating set polytope of trees
Given a graph G and a nonnegative integer number k, a k- dominating set in G is a subset of vertices D such that every vertex in the graph is adjacent to at least k elements of D. The k-dominating set polytope is the convex hull of the incidence vectors of k-dominating sets in G. This is a natural generalization of the well-known dominating set polytope in graphs. In this work we study the 2-dominating set polytope of trees and we will provide new facet de ning inequalities for it.Sociedad Argentina de Informática e Investigación Operativ
New facets of the 2-dominating set polytope of trees
Given a graph G and a nonnegative integer number k, a k- dominating set in G is a subset of vertices D such that every vertex in the graph is adjacent to at least k elements of D. The k-dominating set polytope is the convex hull of the incidence vectors of k-dominating sets in G. This is a natural generalization of the well-known dominating set polytope in graphs. In this work we study the 2-dominating set polytope of trees and we will provide new facet de ning inequalities for it.Sociedad Argentina de Informática e Investigación Operativ
New facets of the 2-dominating set polytope of trees
Given a graph G and a nonnegative integer number k, a k- dominating set in G is a subset of vertices D such that every vertex in the graph is adjacent to at least k elements of D. The k-dominating set polytope is the convex hull of the incidence vectors of k-dominating sets in G. This is a natural generalization of the well-known dominating set polytope in graphs. In this work we study the 2-dominating set polytope of trees and we will provide new facet de ning inequalities for it.Sociedad Argentina de Informática e Investigación Operativ
Limited packings: related vertex partitions and duality issues
A -limited packing partition (LP partition) of a graph is a
partition of into -limited packing sets. We consider the LP
partitions with minimum cardinality (with emphasis on ). The minimum
cardinality is called LP partition number of and denoted by
. This problem is the dual problem of -tuple domatic
partitioning as well as a generalization of the well-studied -distance
coloring problem in graphs.
We give the exact value of for trees and bound it for
general graphs. A section of this paper is devoted to the dual of this problem,
where we give a solution to an open problem posed in . We also revisit
the total limited packing number in this paper and prove that the problem of
computing this parameter is NP-hard even for some special families of graphs.
We give some inequalities concerning this parameter and discuss the difference
between TLP number and LP number with emphasis on trees