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| Natura: | Preprint |
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2023
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| Accesso online: | https://arxiv.org/abs/2308.15603 |
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| _version_ | 1866909162134306816 |
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| author | Cornet, María Gracia Torres, Pablo |
| author_facet | Cornet, María Gracia Torres, Pablo |
| contents | This paper considers multiple domination on Kneser graphs. We focus on $k$-tuple dominating sets, $2$-packings and the associated graph parameters $k$-tuple domination number and $2$-packing number. In particular, we compute the $2$-packing number of Kneser graphs $K(3r-2,r)$ and in odd graphs we obtain minimum $k$-tuple dominating sets of $K(7,3)$ and $K(11,5)$ for every $k$. Besides, we determine the Kneser graphs $K(n,r)$ with $k$-tuple domination number exactly $k+r$ and find all the minimum $k$-tuple dominating sets for these graphs, which generalize results for domination on Kneser graphs. Finally, we give a characterization of the $k$-tuple dominating sets of $K(n,2)$ in terms of the occurrences of the elements in $[n]$, which allows us to obtain minimum sized $k$-tuple dominating sets of $K(n,2)$ for $n\geq Ω(\sqrt{k})$.
Keywords: Kneser graphs, multiple domination, $k$-tuple domination, $2$-packings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_15603 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | $k$-tuple domination on Kneser graphs Cornet, María Gracia Torres, Pablo Combinatorics This paper considers multiple domination on Kneser graphs. We focus on $k$-tuple dominating sets, $2$-packings and the associated graph parameters $k$-tuple domination number and $2$-packing number. In particular, we compute the $2$-packing number of Kneser graphs $K(3r-2,r)$ and in odd graphs we obtain minimum $k$-tuple dominating sets of $K(7,3)$ and $K(11,5)$ for every $k$. Besides, we determine the Kneser graphs $K(n,r)$ with $k$-tuple domination number exactly $k+r$ and find all the minimum $k$-tuple dominating sets for these graphs, which generalize results for domination on Kneser graphs. Finally, we give a characterization of the $k$-tuple dominating sets of $K(n,2)$ in terms of the occurrences of the elements in $[n]$, which allows us to obtain minimum sized $k$-tuple dominating sets of $K(n,2)$ for $n\geq Ω(\sqrt{k})$. Keywords: Kneser graphs, multiple domination, $k$-tuple domination, $2$-packings. |
| title | $k$-tuple domination on Kneser graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2308.15603 |