{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,13]],"date-time":"2026-02-13T10:32:09Z","timestamp":1770978729393,"version":"3.50.1"},"reference-count":11,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2020,12,11]],"date-time":"2020-12-11T00:00:00Z","timestamp":1607644800000},"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>Transitivity on graphs is a concept widely investigated. This suggest to analyze the action of automorphisms on other sets. In this paper, we study the action on the family of \u03b3-sets (minimum dominating sets), the graph is called \u03b3-transitive if given two \u03b3-sets there exists an automorphism which maps one onto the other. We deal with two families: paths Pn and cycles Cn. Their \u03b3-sets are fully characterized and the action of the automorphism group on the family of \u03b3-sets is fully analyzed.<\/jats:p>","DOI":"10.3390\/sym12122053","type":"journal-article","created":{"date-parts":[[2020,12,10]],"date-time":"2020-12-10T22:15:36Z","timestamp":1607638536000},"page":"2053","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Transitivity on Minimum Dominating Sets of Paths and Cycles"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1429-3644","authenticated-orcid":false,"given":"Juan C.","family":"Hern\u00e1ndez-G\u00f3mez","sequence":"first","affiliation":[{"name":"Faculty of Mathematics, Autonomous University of Guerrero, Carlos E. Adame 5, Col. La Garita, 39087 Acapulco, Guerrero, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Gerardo","family":"Reyna-H\u00e9rnandez","sequence":"additional","affiliation":[{"name":"Faculty of Mathematics, Autonomous University of Guerrero, Carlos E. Adame 5, Col. La Garita, 39087 Acapulco, Guerrero, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jes\u00fas","family":"Romero-Valencia","sequence":"additional","affiliation":[{"name":"Faculty of Mathematics, Autonomous University of Guerrero, Sauce 19, La Cima, 39086 Chilpancingo de los Bravo, Guerrero, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8909-1978","authenticated-orcid":false,"given":"Omar","family":"Rosario Cayetano","sequence":"additional","affiliation":[{"name":"Faculty of Mathematics, Autonomous University of Guerrero, Carlos E. Adame 5, Col. La Garita, 39087 Acapulco, Guerrero, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,12,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1108","DOI":"10.1016\/j.disc.2018.12.020","article-title":"New families of graphs determined by their generalized spectrum","volume":"342","author":"Fenjin","year":"2019","journal-title":"Discret. Math."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1141","DOI":"10.1080\/09720529.2019.1689612","article-title":"Chromatic number of some families of graphs","volume":"22","author":"Rani","year":"2019","journal-title":"J. Discret. Math. Sci. Cryptogr."},{"key":"ref_3","first-page":"41","article-title":"Puttaswamy. Transversal domination in graphs","volume":"6","author":"Nayaka","year":"2018","journal-title":"Gulf J. Math."},{"key":"ref_4","unstructured":"Garey, M.R., and Johnson, D.S. (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness, W.H. Freeman and Company."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1007\/BF01848079","article-title":"On graphs having domination number half their order","volume":"16","author":"Fink","year":"1985","journal-title":"Period. Math. Hung."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1016\/0012-365X(90)90114-W","article-title":"Estimations for the domination number of a graph","volume":"80","author":"Flach","year":"1990","journal-title":"Discret. Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/0012-365X(91)90071-9","article-title":"Maximum number of edges in connected graphs with a given domination number","volume":"87","author":"Sanchis","year":"1991","journal-title":"Discret. Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1016\/j.dam.2014.12.010","article-title":"Enumerating minimal dominating sets in chordal bipartite graphs","volume":"199","author":"Golovach","year":"2016","journal-title":"Discret. Appl. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"301","DOI":"10.7151\/dmgt.2016","article-title":"On the total k\u2014domination in graphs","volume":"38","author":"Bermudo","year":"2018","journal-title":"Discuss. Math. Graph Theory"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1016\/j.dam.2018.05.025","article-title":"On the global total k-domination number of graphs","volume":"263","author":"Bermudo","year":"2019","journal-title":"Discret. Appl. Math."},{"key":"ref_11","first-page":"425","article-title":"\u03b3\u2014transitivity in Platonic graphs","volume":"18","year":"2017","journal-title":"Adv. Appl. Discret. Math."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/12\/2053\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:43:37Z","timestamp":1760179417000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/12\/2053"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,12,11]]},"references-count":11,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2020,12]]}},"alternative-id":["sym12122053"],"URL":"https:\/\/doi.org\/10.3390\/sym12122053","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,12,11]]}}}