Article in volume
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Title:
Further results on packing related parameters in graphs
PDFSource:
Discussiones Mathematicae Graph Theory 42(2) (2022) 333-348
Received: 2018-11-19 , Revised: 2019-08-28 , Accepted: 2019-10-17 , Available online: 2019-11-26 , https://doi.org/10.7151/dmgt.2262
Abstract:
Given a graph $G=(V,E)$, a set $B\subseteq V(G)$ 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 $n\geq 2$ improving the upper bound
$\gamma_{t}(T)\leq(n+s)/2$ given by Chellali and Haynes in 2004, in which $s$
is the number of support vertices of $T$.
Keywords:
packing number, open packing number, independence number, Nordhaus-Gaddum inequality, total domination number
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