A set B of vertices in a graph G is called a k-limited packing if for each vertex v of G, its closed neighbourhood has at most k vertices in B. The k-limited packing number of a graph G, denoted by \(L_k(G)\), is the largest number of vertices in a k-limited packing in G. The concept of the k-limited packing of a graph was introduced by Gallant et al., which is a generalization of the well-known packing of a graph. In this paper, we present some tight bounds for the k-limited packing number of a graph in terms of its order, diameter, girth, and maximum degree, respectively. As a result, we obtain a tight Nordhaus–Gaddum type result for the k-limited packing number. At last, we investigate the relationship among the open packing number, the packing number and 2-limited packing number of trees.
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Balister PN, Bollobás B, Gunderson K (2015) Limited packings of closed neighbourhoods in graphs. arXiv:1501.01833 [math.CO]
Berge C (1973) Graphs and hypergraphs. North Holland, Amsterdam
Biggs N (1973) Perfect codes in graphs. J Comb Theory Ser B 15:289–296
Bondy JA, Murty USR (2008) Graph theory, GTM 224. Springer, Berlin
Clark L (1993) Perfect domination in random graphs. J Comb Math Comb Comput 14:173–182
Cockayne EJ, Dawes RM, Hedetniemi ST (1980) Total domination in graphs. Networks 10:211–219
Dobson MP, Leoni V, Nasini G (2011) The multiple domination and limited packing problems in graphs. Inf Process Lett 111:1108–1113
Favaron O, Henning MA (2003) Upper total domination in claw-free graphs. J Graph Theory 44:148–158
Favaron O, Henning MA, Mynhardt CM, Puech J (2000) Total domination in graphs with minimum degree three. J Graph Theory 34:9–19
Gagarin A, Zverovich V (2015) The probabilistic approach to limited packings in graphs. Discrete Appl Math 184:146–153
Gallant R, Gunther G, Hartnell BL, Rall DF (2010) Limited packings in graphs. Discrete Appl Math 158:1357–1364
Hamid IS, Saravanakumar S (2015) Packing parameters in graphs. Discuss Math Gaph Theory 35:5–16
Haynes TW, Hedetniemi ST, Slater PJ (1998a) Fundamentals of domination in graphs. Marcel Dekker, New York
Haynes TW, Hedetniemi ST, Slater PJ (1998b) Domination in graphs: advanced topics. Marcel Dekker, New York
Henning MA (1998) Packing in trees. Discrete Math 186:145–155
Henning MA, Slater PJ (1999) Open packing in graphs. J Comb Math Comb Comput 28:5–18
Henning MA, Yeo A (2007) A new upper bound on the total domination number of a graph. Electron J Comb 14:R65
Hochbaum DS, Schmoys DB (1985) A best possible heuristic for the \(k\)-center problem. Math Oper Res 10:180–184
Jaeger F, Payan C (1972) Relations du type Nordhaus–Gaddum pour le nombre dábsorption dún graphe simple. C R Acad Sci Paris A 274:728–730
Leoni V, Nasini G (2014) Limited packing and multiple domination problems: polynomial time reductions. Discrete Appl Math 164:547–553
Meir A, Moon JW (1975) Relations between packing and covering numbers of a tree. Pac J Math 61:225–233
Mojdeh DA, Samadi B (2019) Packing parameters in graphs: new bounds and a solution to an open problem. arXiv:1705.08667 [math.CO]
Mojdeh DA, Samadi B, Hosseini Moghaddam SM (2017) Limited packing vs. tuple domination in graphs. Ars Comb 133:155–161
Nordhaus EA, Gauddum JW (1956) On complementary graphs. Am Math Mon 63:175–177
Ore O (1962) Theory of graphs. American Mathematical Society, Providence
Rall DF (2005) Total domination in categorical products of graphs. Discuss Math Graph Theory 25:35–44
Samadi B (2016) On the \(k\)-limited packing numbers in graphs. Discrete Optim 22:270–276
Topp J, Volkmann L (1991) On packing and covering numbers of graphs. Discrete Math 96:229–238
Yeo A (2007) Relationships between total domination, order, size, and maximum degree of graphs. J Graph Theory 55:325–337
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Bai, X., Chang, H. & Li, X. More on limited packings in graphs. J Comb Optim 40, 412–430 (2020). https://doi.org/10.1007/s10878-020-00606-z
- k-limited packing
- Opening packing
- Nordhaus–Gaddum type result
Mathematics Subject Classification