Geodesic
Further Results on Packing Related Parameters in Graphs
Discussiones Mathematicae. Graph Theory, Tome 42 (2022) no. 2, pp. 333-348.

Voir la notice de l'article provenant de la source Library of Science

Given a graph G = (V, E), a set B ⊆ 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 ρ(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 γt(T) for a tree T of order n ≥ 2 improving the upper bound γt(T) ≤ (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 
@article{DMGT_2022_42_2_a1,
     author = {Mojdeh, Doost Ali and Samadi, Babak and Yero, Ismael G.},
     title = {Further {Results} on {Packing} {Related} {Parameters} in {Graphs}},
     journal = {Discussiones Mathematicae. Graph Theory},
     pages = {333--348},
     publisher = {mathdoc},
     volume = {42},
     number = {2},
     year = {2022},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMGT_2022_42_2_a1/}
}
TY  - JOUR
AU  - Mojdeh, Doost Ali
AU  - Samadi, Babak
AU  - Yero, Ismael G.
TI  - Further Results on Packing Related Parameters in Graphs
JO  - Discussiones Mathematicae. Graph Theory
PY  - 2022
SP  - 333
EP  - 348
VL  - 42
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DMGT_2022_42_2_a1/
LA  - en
ID  - DMGT_2022_42_2_a1
ER  - 
%0 Journal Article
%A Mojdeh, Doost Ali
%A Samadi, Babak
%A Yero, Ismael G.
%T Further Results on Packing Related Parameters in Graphs
%J Discussiones Mathematicae. Graph Theory
%D 2022
%P 333-348
%V 42
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DMGT_2022_42_2_a1/
%G en
%F DMGT_2022_42_2_a1
Mojdeh, Doost Ali; Samadi, Babak; Yero, Ismael G. Further Results on Packing Related Parameters in Graphs. Discussiones Mathematicae. Graph Theory, Tome 42 (2022) no. 2, pp. 333-348. http://geodesic.mathdoc.fr/item/DMGT_2022_42_2_a1/

[1] M. Aouchiche and P. Hansen, A survey of Nordhaus-Gaddum type relations, Discrete Appl. Math. 161 (2013) 466–546. https://doi.org/10.1016/j.dam.2011.12.018

[2] M. Borowiecki, On a minimaximal kernel of trees, Discuss. Math. 1 (1975) 3–6.

[3] M. Chellali and T.W. Haynes, A note on the total domination number of a tree, J. Combin. Math. Combin. Comput. 58 (2006) 189–193.

[4] M. Chellali and T.W. Haynes, Total and paired-domination numbers of a tree, AKCE Int. J. Graphs Comb. 1 (2004) 69–75.

[5] E.J. Cockayne, R.M. Dawes and S.T. Hedetniemi, Total domination in graphs, Networks 10 (1980) 211–219. https://doi.org/10.1002/net.3230100304

[6] E.J. Cockayne, P.A. Dreyer, S.M. Hedetniemi and S.T. Hedetniemi, Roman domination in graphs, Discrete Math. 278 (2004) 11–22. https://doi.org/10.1016/j.disc.2003.06.004

[7] W.J. Desormeaux and M.A. Henning, A new lower bound on the total domination number of a tree, Ars Combin. 138 (2018) 305–322.

[8] R. Gallant, G. Gunther, B.L. Hartnell and D.F. Rall, Limited packing in graphs, Discrete Appl. Math. 158 (2010) 1357–1364. https://doi.org/10.1016/j.dam.2009.04.014

[9] M. Gentner and D. Rautenbach, Some comments on the Slater number, Discrete Math. 340 (2017) 1497–1502. https://doi.org/10.1016/j.disc.2017.02.009

[10] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, New York, 1998).

[11] M.A. Henning and P.J. Slater, Open packing in graphs, J. Combin. Math. Combin. Comput. 28 (1999) 5–18.

[12] M.A. Henning and A. Yeo, Total Domination in Graphs (Springer-Verlag, New York, 2013). https://doi.org/10.1007/978-1-4614-6525-6

[13] C.X. Kang, On domination number and distance in graphs, Discrete Appl. Math. 200 (2016) 203–206. https://doi.org/10.1016/j.dam.2015.07.009

[14] A. Meir and J.W. Moon, Relations between packing and covering numbers of a tree, Pacific. J. Math. 61 (1975) 225–233. https://doi.org/10.2140/pjm.1975.61.225

[15] D.A. Mojdeh and B. Samadi, Packing parameters in graphs: new bounds and a solution to an open problem, J. Comb. Optim. 38 (2019) 739–747. https://doi.org/10.1007/s10878-019-00410-4

[16] E.A. Nordhaus and J. Gaddum, On complementary graphs, Amer. Math. Monthly 63 (1956) 175–177. https://doi.org/10.2307/2306658

[17] O. Ore, Theory of Graphs (Amer. Math. Soc. Colloq. Publ. 38, American Mathematical Society, Providence, RI, 1962). https://doi.org/10.1090/coll/038

[18] I. Sahul Hamid and S. Saravanakumar, Packing parameters in graphs, Discuss. Math. Graph Theory 35 (2015) 5–16. https://doi.org/10.7151/dmgt.1775

[19] I. Sahul Hamid and S. Saravanakumar, On open packing number of graphs, Iran. J. Math. Sci. Inform. 12 (2017) 107–117.

[20] D.F. Rall, Total domination in categorical products of graphs, Discuss. Math. Graph Theory 25 (2005) 35–44. https://doi.org/10.7151/dmgt.1257

[21] B. Samadi, On the k-limited packing numbers in graphs, Discrete Optim. 22 (2016) 270–276. https://doi.org/10.1016/j.disopt.2016.08.002

[22] I. Stewart, Defend the Roman Empire!, Sci. Amer. 281 (1999) 136–138. https://doi.org/10.1038/scientificamerican1299-136

[23] D.B. West, Introduction to Graph Theory, Second Edition (Prentice Hall, USA, 2001).