{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:22:05Z","timestamp":1760059325725,"version":"build-2065373602"},"reference-count":21,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2025,6,5]],"date-time":"2025-06-05T00:00:00Z","timestamp":1749081600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004386","name":"University Malaya","doi-asserted-by":"publisher","award":["BKP-ECRG-BKP007-2025-ECRG"],"award-info":[{"award-number":["BKP-ECRG-BKP007-2025-ECRG"]}],"id":[{"id":"10.13039\/501100004386","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Let G be a group and let A1,A2 be nonempty subsets of G. We say that [A1,A2] is a 2-complete decomposition of G if G=A1A2 and A1,A2 are disjoint. In this paper, we define a factor graph of a group and determine the symmetrical relationship between factor graphs and 2-complete decompositions. We then characterize groups that have a complete decomposition [A1,A2] where one of the sets, A1,A2, is of size two, or of size three and contains the identity.<\/jats:p>","DOI":"10.3390\/sym17060883","type":"journal-article","created":{"date-parts":[[2025,6,5]],"date-time":"2025-06-05T05:49:22Z","timestamp":1749102562000},"page":"883","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Graphical Approach: 2-Complete Decompositions of Groups"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0009-0001-1138-5714","authenticated-orcid":false,"given":"Lei","family":"Dong","sequence":"first","affiliation":[{"name":"Faculty of Applied Science, Macao Polytechnic University, Macau 999078, China"},{"name":"Institute of Mathematical Sciences, Faculty of Science, Universiti Malaya, Kuala Lumpur 50603, Malaysia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3554-322X","authenticated-orcid":false,"given":"Kah Lun","family":"Wang","sequence":"additional","affiliation":[{"name":"Institute of Mathematical Sciences, Faculty of Science, Universiti Malaya, Kuala Lumpur 50603, Malaysia"},{"name":"Centre of Research for Statistical Modelling and Methodology, Faculty of Science, Universiti Malaya, Kuala Lumpur 50603, Malaysia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kok Bin","family":"Wong","sequence":"additional","affiliation":[{"name":"Institute of Mathematical Sciences, Faculty of Science, Universiti Malaya, Kuala Lumpur 50603, Malaysia"},{"name":"Centre of Research for Statistical Modelling and Methodology, Faculty of Science, Universiti Malaya, Kuala Lumpur 50603, Malaysia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,6,5]]},"reference":[{"key":"ref_1","first-page":"171","article-title":"Covering multidimensional spaces by cube lattices","volume":"45","year":"1938","journal-title":"Mat. 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