{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,8,8]],"date-time":"2024-08-08T05:46:56Z","timestamp":1723096016547},"reference-count":0,"publisher":"Combinatorial Press","issue":"01","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Ars Comb."],"published-print":{"date-parts":[[2024,1]]},"abstract":"<jats:p>A linear system is a pair \\((P,\\mathcal{L})\\) where \\(\\mathcal{L}\\) is a finite family of subsets on a finite ground set \\(P\\) such that any two subsets of \\(\\mathcal{L}\\) share at most one element. Furthermore, if for every two subsets of \\(\\mathcal{L}\\) share exactly one element, the linear system is called intersecting. A linear system \\((P,\\mathcal{L})\\) has rank \\(r\\) if the maximum size of any element of \\(\\mathcal{L}\\) is \\(r\\). By \\(\\gamma(P,\\mathcal{L})\\) and \\(\\nu_2(P,\\mathcal{L})\\) we denote the size of the minimum dominating set and the maximum 2-packing of a linear system \\((P,\\mathcal{L})\\), respectively. It is known that any intersecting linear system \\((P,\\mathcal{L})\\) of rank \\(r\\) is such that \\(\\gamma(P,\\mathcal{L})\\leq r-1\\). Li et al. in [S. Li, L. Kang, E. Shan and Y. Dong, The finite projective plane and the 5-Uniform linear intersecting hypergraphs with domination number four, Graphs and 34 Combinatorics (2018) , no.~5, 931\u2013945.] proved that every intersecting linear system of rank 5 satisfying \\(\\gamma(P,\\mathcal{L})=4\\) can be constructed from a 4-uniform intersecting linear subsystem \\((P^\\prime,\\mathcal{L}^\\prime)\\) of the projective plane of order 3 satisfying \\(\\tau(P^\\prime,\\mathcal{L}^\\prime)=\\nu_2(P^\\prime,\\mathcal{L}^\\prime)=4\\), where \\(\\tau(P^\\prime,\\mathcal{L}^\\prime)\\) is the transversal number of \\((P^\\prime,\\mathcal{L}^\\prime)\\). In this paper, we give an alternative proof of this result given by Li et al., giving a complete characterization of these 4-uniform intersecting linear subsystems. Moreover, we prove a general case, that is, we prove if $q$ is an odd prime power and \\((P,\\mathcal{L})\\) is an intersecting linear system of rank \\((q+2)\\) satisfying \\(\\gamma(P,\\mathcal{L})=q+1\\), then this linear system can be constructed from a spanning \\((q+1)\\)-uniform intersecting linear subsystem \\((P^\\prime,\\mathcal{L}^\\prime)\\) of the projective plane of order \\(q\\) satisfying \\(\\tau(P^\\prime,\\mathcal{L}^\\prime)=\\nu_2(P^\\prime,\\mathcal{L}^\\prime)=q+1\\).<\/jats:p>","DOI":"10.61091\/ars158-05","type":"journal-article","created":{"date-parts":[[2024,4,14]],"date-time":"2024-04-14T22:33:20Z","timestamp":1713134000000},"page":"35-40","source":"Crossref","is-referenced-by-count":0,"title":["On Domination and 2-packing Numbers in Intersecting Linear Systems"],"prefix":"10.61091","volume":"158","author":[{"name":"Subdirecci\u00f3n de Ingenier\u00eda y Posgrado Universidad Aeron\u00e1utica en Quer\u00e9taro Parque Aeroespacial de Quer\u00e9taro 76278, Quer\u00e9taro, M\u00e9xico","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Adri\u00e1n V\u00e1zquez","family":"\u00c1vila","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"39747","container-title":["Ars Combinatoria"],"original-title":[],"deposited":{"date-parts":[[2024,4,14]],"date-time":"2024-04-14T22:33:27Z","timestamp":1713134007000},"score":1,"resource":{"primary":{"URL":"https:\/\/combinatorialpress.com\/ars-articles\/volume-158\/on-domination-and-2-packing-numbers-in-intersecting-linear-systems\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,1]]},"references-count":0,"journal-issue":{"issue":"01","published-online":{"date-parts":[[2024,3,31]]},"published-print":{"date-parts":[[2024,3,31]]}},"URL":"https:\/\/doi.org\/10.61091\/ars158-05","relation":{},"ISSN":["0381-7032","2817-5204"],"issn-type":[{"type":"print","value":"0381-7032"},{"type":"electronic","value":"2817-5204"}],"subject":[],"published":{"date-parts":[[2024,1]]}}}