{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,20]],"date-time":"2026-05-20T19:50:29Z","timestamp":1779306629622,"version":"3.51.4"},"reference-count":31,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2021,12,12]],"date-time":"2021-12-12T00:00:00Z","timestamp":1639267200000},"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 2008, Hedetniemi et al. introduced (1,k)-domination in graphs. The research on this concept was extended to the problem of existence of independent (1,k)-dominating sets, which is an NP-complete problem. In this paper, we consider independent (1,1)- and (1,2)-dominating sets, which we name as (1,1)-kernels and (1,2)-kernels, respectively. We obtain a complete characterization of generalized corona of graphs and G-join of graphs, which have such kernels. Moreover, we determine some graph parameters related to these sets, such as the number and the cardinality. In general, graph products considered in this paper have an asymmetric structure, contrary to other many well-known graph products (Cartesian, tensor, strong).<\/jats:p>","DOI":"10.3390\/sym13122399","type":"journal-article","created":{"date-parts":[[2021,12,13]],"date-time":"2021-12-13T01:29:33Z","timestamp":1639358973000},"page":"2399","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["On Independent Secondary Dominating Sets in Generalized Graph Products"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8776-5270","authenticated-orcid":false,"given":"Adrian","family":"Michalski","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"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1629-0167","authenticated-orcid":false,"given":"Pawe\u0142","family":"Bednarz","sequence":"additional","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"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,12,12]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Diestel, R. (2005). Graph Theory, Springer. [3rd ed.].","DOI":"10.4171\/owr\/2005\/03"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"118","DOI":"10.1016\/j.dam.2018.03.082","article-title":"On the super domination number of lexicographic product graphs","volume":"263","author":"Dettlaff","year":"2019","journal-title":"Discret. Appl. 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