{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,14]],"date-time":"2026-02-14T02:32:33Z","timestamp":1771036353763,"version":"3.50.1"},"reference-count":20,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2025,3,8]],"date-time":"2025-03-08T00:00:00Z","timestamp":1741392000000},"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 this paper, we investigate the existence of perfect (1,2)-dominating sets ((1,2)-PDS) in graphs with three vertices of maximum degree equal to n\u22122. A perfect (1,2)-dominating set is a special case of a (1,2)-dominating set. Graphs with such dominating sets may exhibit a symmetric structure. If a graph has at least one vertex of degree one, then it has a (1,2)-PDS. Hence, we consider only graphs with a minimum degree greater than or equal to 2. Therefore, a symmetric or asymmetric structure of graphs can be useful in determining whether a graph has a (1,2)-PDS. On the other hand, the symmetric or asymmetric structure may be even more helpful when studying the existence of (1,2)-PDS in relation to the maximum degree of a graph. Moreover, we analyze the structural conditions under which (1,2)-PDS exist, considering cases where the three vertices are adjacent or nonadjacent, and whether their neighborhoods are identical or distinct. Our study provides necessary and sufficient conditions for the existence of (1,2)-PDS in the given cases, extending the understanding of (1,2)-domination and its applications.<\/jats:p>","DOI":"10.3390\/sym17030405","type":"journal-article","created":{"date-parts":[[2025,3,10]],"date-time":"2025-03-10T08:46:41Z","timestamp":1741596401000},"page":"405","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["On the Existence of Perfect (1, 2)-Dominating Sets in Graphs"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4856-9613","authenticated-orcid":false,"given":"Urszula","family":"Bednarz","sequence":"first","affiliation":[{"name":"The Faculty of Mathematics and Applied Physics, Rzesz\u00f3w University of Technology, al. Powsta\u0144c\u00f3w Warszawy 12, 35-959 Rzesz\u00f3w, Poland"}]}],"member":"1968","published-online":{"date-parts":[[2025,3,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Diestel, R. (2005). Graph Theory, Springer.","DOI":"10.4171\/owr\/2005\/03"},{"key":"ref_2","unstructured":"de Jaenisch, C.F. (1862). Traite des Applications de L\u2019Analyse Mathematique au Jeu des Echecs, Academie Imperialedes Sciences."},{"key":"ref_3","unstructured":"Berge, C. (1962). The Theory of Graphs and Its Applications, Wiley."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Ore, O. (1962). Theory of Graphs, American Mathematical Society.","DOI":"10.1090\/coll\/038"},{"key":"ref_5","first-page":"117","article-title":"Secondary domination in graph","volume":"5","author":"Hedetniemi","year":"2008","journal-title":"AKCE Int. J. Graphs Comb."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1016\/j.dam.2023.06.021","article-title":"Double domination in rooted product graphs","volume":"339","year":"2023","journal-title":"Discrete Appl. Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2277","DOI":"10.1016\/j.disc.2007.04.057","article-title":"On graphs with equal domination and 2-domination numbers","volume":"308","author":"Hansberg","year":"2008","journal-title":"Discrete Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"29","DOI":"10.7151\/dmgt.1256","article-title":"On double domination in graphs","volume":"25","author":"Harant","year":"2005","journal-title":"Discuss. Math. Graph Theory"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"268","DOI":"10.1016\/j.dam.2022.01.002","article-title":"Minimum 2-dominating sets in regular graphs","volume":"323","author":"Hoppen","year":"2022","journal-title":"Discrete Appl. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"7050","DOI":"10.1002\/mma.8223","article-title":"On proper (1, 2)-dominating sets in graphs","volume":"45","author":"Michalski","year":"2022","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Arumugam, S., Bagga, J., Beineke, L., and Panda, B. (2017). (1, 2)-Domination in Graphs. Theoretical Computer Science and Discrete Mathematics, Springer.","DOI":"10.1007\/978-3-319-64419-6"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Raczek, J. (2022). Polynomial Algorithm for Minimal (1, 2)-Dominating Set in Networks. Electronics, 11.","DOI":"10.3390\/electronics11030300"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1163","DOI":"10.1007\/s00453-023-01192-2","article-title":"Complexity Issues on of Secondary Domination Number","volume":"86","author":"Raczek","year":"2024","journal-title":"Algorithmica"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Michalski, A., and Bednarz, P. (2021). On Independent Secondary Dominating Sets in Generalized Graph Products. Symmetry, 13.","DOI":"10.3390\/sym13122399"},{"key":"ref_15","first-page":"125155","article-title":"On the existence and the number of independent (1, 2)- dominating sets in the G-join of graphs","volume":"377","author":"Michalski","year":"2020","journal-title":"Appl. Math. Comput."},{"key":"ref_16","unstructured":"Michalski, A. (2019). Secondary dominating sets in graphs and their modification. Book of Abstracts, The 7th Gda\u0144sk Workshop on Graph Theory, Gda\u0144sk, Poland, 1\u20135 July 2019, The University of Gda\u0144sk."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"813","DOI":"10.7494\/OpMath.2023.43.6.813","article-title":"On minimum intersections of certain secondary dominating sets in graphs","volume":"43","author":"Kosiorowska","year":"2023","journal-title":"Opuscula Math."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Bednarz, P., and Pirga, M. (2024). On Proper 2-Dominating Sets in Graphs. Symmetry, 16.","DOI":"10.3390\/sym16030296"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"53","DOI":"10.7494\/OpMath.2025.45.1.53","article-title":"(1, 2)-PDS in graphs with the small number of vertices of large degrees","volume":"45","author":"Bednarz","year":"2025","journal-title":"Opuscula Math."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Cornejo Acosta, J.A., Garcia D\u00edaz, J., Menchaca-M\u00e9ndez, R., and Menchaca-M\u00e9ndez, R. (2020). Solving the Capacitated Vertex K-Center Problem through the Minimum Capacitated Dominating Set Problem. Mathematics, 8.","DOI":"10.3390\/math8091551"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/3\/405\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T16:49:25Z","timestamp":1760028565000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/3\/405"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,3,8]]},"references-count":20,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2025,3]]}},"alternative-id":["sym17030405"],"URL":"https:\/\/doi.org\/10.3390\/sym17030405","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,3,8]]}}}