9,787 research outputs found
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
Locating-dominating sets and identifying codes in graphs of girth at least 5
Locating-dominating sets and identifying codes are two closely related
notions in the area of separating systems. Roughly speaking, they consist in a
dominating set of a graph such that every vertex is uniquely identified by its
neighbourhood within the dominating set. In this paper, we study the size of a
smallest locating-dominating set or identifying code for graphs of girth at
least 5 and of given minimum degree. We use the technique of vertex-disjoint
paths to provide upper bounds on the minimum size of such sets, and construct
graphs who come close to meet these bounds.Comment: 20 pages, 9 figure
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