Further results on packing related parameters in graphs

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URI: http://hdl.handle.net/10498/26650
DOI: 10.7151/dmgt.2262
ISSN: 1234-3099
ISSN: 2083-5892
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2022-05Department
MatemáticasSource
Discussiones Mathematicae - Graph Theory, Vol. 42, Núm. 2, pp. 333-348Abstract
Given a graph G = (V, E), a set B subset of V (G) is a packing in G if the closed neighborhoods of every pair of distinct vertices in B are pairwise disjoint. The packing number rho(G) of G is the maximum cardinality of a packing in G. Similarly, open packing sets and open packing number are defined for a graph G by using open neighborhoods instead of closed ones. We give several results concerning the (open) packing number of graphs in this paper. For instance, several bounds on these packing parameters along with some Nordhaus-Gaddum inequalities are given. We characterize all graphs with equal packing and independence numbers and give the characterization of all graphs for which the packing number is equal to the independence number minus one. In addition, due to the close connection between the open packing and total domination numbers, we prove a new upper bound on the total domination number gamma(t)(T) for a tree T of order n >= 2 improving the upper bound gamma(t)(T) <= (n + s)/2 given by Chellali and Haynes in 2004, in which s is the number of support vertices of T.
Subjects
packing number; open packing number; independence number; Nordhaus-Gaddum inequality; total domination numberCollections
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