{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:56:56Z","timestamp":1760144216898,"version":"build-2065373602"},"reference-count":16,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2024,3,31]],"date-time":"2024-03-31T00:00:00Z","timestamp":1711843200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12261086"],"award-info":[{"award-number":["12261086"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The restricted edge-connectivity of a connected graph G, denoted by \u03bb\u2032(G), if it exists, is the minimum cardinality of a set of edges whose deletion makes G disconnected, and each component has at least two vertices. It was proved that \u03bb\u2032(G) exists if and only if G has at least four vertices and G is not a star. In this case, a graph G is called maximally restricted edge-connected if \u03bb\u2032(G)=\u03be(G), and a graph G is called super restricted edge-connected if each minimum restricted edge-cut isolates an edge of G. The strong product of graphs G and H, denoted by G\u22a0H, is the graph with the vertex set V(G)\u00d7V(H) and the edge set {(x1,y1)(x2,y2)|x1=x2 and y1y2\u2208E(H); or y1=y2 and x1x2\u2208E(G); or x1x2\u2208E(G) and y1y2\u2208E(H)}. In this paper, we determine, for any nontrivial connected graph G, the restricted edge-connectivity of G\u22a0Pn, G\u22a0Cn and G\u22a0Kn, where Pn, Cn and Kn are the path, cycle and complete graph of order n, respectively. As corollaries, we give sufficient conditions for these strong product graphs G\u22a0Pn, G\u22a0Cn and G\u22a0Kn to be maximally restricted edge-connected and super restricted edge-connected.<\/jats:p>","DOI":"10.3390\/axioms13040231","type":"journal-article","created":{"date-parts":[[2024,3,31]],"date-time":"2024-03-31T13:32:56Z","timestamp":1711891976000},"page":"231","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The Restricted Edge-Connectivity of Strong Product Graphs"],"prefix":"10.3390","volume":"13","author":[{"given":"Hazhe","family":"Ye","sequence":"first","affiliation":[{"name":"College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1595-9834","authenticated-orcid":false,"given":"Yingzhi","family":"Tian","sequence":"additional","affiliation":[{"name":"College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,3,31]]},"reference":[{"key":"ref_1","unstructured":"Bondy, J.A., and Murty, U.S.R. (2008). Graduate Texts in Mathematics, Springer."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1016\/0020-0190(88)90025-7","article-title":"On computing a conditional edge-connectivity of a graph","volume":"27","author":"Esfahanian","year":"1988","journal-title":"Inf. Process. Lett."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1142\/S1793557108000102","article-title":"On the edge-connectivity of Cartesian product graphs","volume":"1","author":"Klavzar","year":"2008","journal-title":"Asian Eur. J. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1002\/net.10037","article-title":"Super edge- and point-connectivities of the Cartesian product of regular graphs","volume":"40","author":"Shieh","year":"2002","journal-title":"Networks"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"655","DOI":"10.1016\/j.ipl.2009.02.025","article-title":"Super restricted edge connected Cartesian product graphs","volume":"109","author":"Liu","year":"2009","journal-title":"Inf. Process. Lett."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"152","DOI":"10.1002\/net.20149","article-title":"On super edge-connectivity of Cartesian product graphs","volume":"49","author":"Chen","year":"2007","journal-title":"Networks"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"478","DOI":"10.1016\/j.disc.2010.12.012","article-title":"On optimizing edge-connectivity of product graphs","volume":"311","author":"Ou","year":"2011","journal-title":"Discret. Math."},{"key":"ref_8","first-page":"45","article-title":"On the connectivity of the direct product of graphs","volume":"41","year":"2008","journal-title":"Australas. J. Combin."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"899","DOI":"10.1016\/j.ipl.2011.06.007","article-title":"On edge connectivity of direct products of graphs","volume":"111","author":"Cao","year":"2011","journal-title":"Inf. Process. Lett."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1385","DOI":"10.1016\/j.disc.2013.02.011","article-title":"A characterization of the edge connectivity of direct products of graphs","volume":"313","year":"2013","journal-title":"Discret. Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1950012","DOI":"10.1142\/S0129626419500129","article-title":"The Restricted Edge-Connectivity of Kronecker Product Graphs","volume":"29","author":"Ma","year":"2019","journal-title":"Parallel Process. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2350008","DOI":"10.1142\/S0129626423500081","article-title":"The Super Restricted Edge-connectedness of Direct Product Graphs","volume":"33","author":"Bai","year":"2023","journal-title":"Parallel Process. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"333","DOI":"10.7151\/dmgt.1365","article-title":"Edge-connectivity of strong products of graphs","volume":"27","year":"2007","journal-title":"Discuss. Math. Graph Theory"},{"key":"ref_14","first-page":"55","article-title":"On restricted edge connectivity of strong product graphs","volume":"123","author":"Ou","year":"2015","journal-title":"Ars Comb."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1750007","DOI":"10.1142\/S0219265917500074","article-title":"Super Edge-Connectivity of Strong Product Graphs","volume":"17","author":"Wang","year":"2017","journal-title":"J. Interconnect. Netw."},{"key":"ref_16","first-page":"449","article-title":"Connectivity and edge-connectivity of strong product graphs","volume":"38","author":"Yang","year":"2008","journal-title":"J. Univ. Sci. Technol. China"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/4\/231\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:21:53Z","timestamp":1760106113000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/4\/231"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,3,31]]},"references-count":16,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2024,4]]}},"alternative-id":["axioms13040231"],"URL":"https:\/\/doi.org\/10.3390\/axioms13040231","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2024,3,31]]}}}