{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T14:12:33Z","timestamp":1760364753402,"version":"build-2065373602"},"reference-count":14,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2020,1,9]],"date-time":"2020-01-09T00:00:00Z","timestamp":1578528000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In the paper, we extend known results concerning crossing numbers of join products of small graphs of order six with discrete graphs. The crossing number of the join product      G \u2217  +  D n      for the graph     G \u2217     on six vertices consists of one vertex which is adjacent with three non-consecutive vertices of the 5-cycle. The proofs were based on the idea of establishing minimum values of crossings between two different subgraphs that cross the edges of the graph     G \u2217     exactly once. These minimum symmetrical values are described in the individual symmetric tables.<\/jats:p>","DOI":"10.3390\/sym12010135","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T04:06:51Z","timestamp":1578629211000},"page":"135","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["On the Crossing Numbers of the Joining of a Specific Graph on Six Vertices with the Discrete Graph"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2837-8879","authenticated-orcid":false,"given":"Michal","family":"Sta\u0161","sequence":"first","affiliation":[{"name":"Faculty of Electrical Engineering and Informatics, Technical University of Ko\u0161ice, 042 00 Ko\u0161ice, Slovakia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,1,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"312","DOI":"10.1137\/0604033","article-title":"Crossing number is NP-complete","volume":"4","author":"Garey","year":"1983","journal-title":"SIAM J. 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Proceedings of the 24th Annual European Symposium on Algorithms (ESA 2016), Aarhus, Denmark."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/1\/135\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T13:29:10Z","timestamp":1760362150000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/1\/135"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,1,9]]},"references-count":14,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2020,1]]}},"alternative-id":["sym12010135"],"URL":"https:\/\/doi.org\/10.3390\/sym12010135","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2020,1,9]]}}}