{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:12:42Z","timestamp":1760235162999,"version":"build-2065373602"},"reference-count":11,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2021,8,2]],"date-time":"2021-08-02T00:00:00Z","timestamp":1627862400000},"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>The cardinality of a largest independent set of G, denoted by \u03b1(G), is called the independence number of G. The independent domination number i(G) of a graph G is the cardinality of a smallest independent dominating set of G. We introduce the concept of the common independence number of a graph G, denoted by \u03b1c(G), as the greatest integer r such that every vertex of G belongs to some independent subset X of VG with |X|\u2265r. The common independence number \u03b1c(G) of G is the limit of symmetry in G with respect to the fact that each vertex of G belongs to an independent set of cardinality \u03b1c(G) in G, and there are vertices in G that do not belong to any larger independent set in G. For any graph G, the relations between above parameters are given by the chain of inequalities i(G)\u2264\u03b1c(G)\u2264\u03b1(G). In this paper, we characterize the trees T for which i(T)=\u03b1c(T), and the block graphs G for which \u03b1c(G)=\u03b1(G).<\/jats:p>","DOI":"10.3390\/sym13081411","type":"journal-article","created":{"date-parts":[[2021,8,2]],"date-time":"2021-08-02T08:44:11Z","timestamp":1627893851000},"page":"1411","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Common Independence in Graphs"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7296-1893","authenticated-orcid":false,"given":"Magda","family":"Dettlaff","sequence":"first","affiliation":[{"name":"Technical Physics and Applied Mathematics Department, Gda\u0144sk University of Technology, 80-803 Gda\u0144sk, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Magdalena","family":"Lema\u0144ska","sequence":"additional","affiliation":[{"name":"Technical Physics and Applied Mathematics Department, Gda\u0144sk University of Technology, 80-803 Gda\u0144sk, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8069-7850","authenticated-orcid":false,"given":"Jerzy","family":"Topp","sequence":"additional","affiliation":[{"name":"Institute of Applied Informatics, The State University of Applied Sciences in Elbla\u0327g, 82-300 Elbla\u0327g, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,8,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Haynes, T.W., Hedetniemi, S.T., and Slater, P.J. (1998). Fundamentals of Domination in Graphs, Marcel Dekker, Inc.","DOI":"10.1002\/(SICI)1097-0037(199810)32:3<199::AID-NET4>3.0.CO;2-F"},{"key":"ref_2","unstructured":"Berge, C. (1962). Theory of Graphs and Its Applications, Methuen."},{"key":"ref_3","unstructured":"Berge, C. (1973). Graphs and Hypergraphs, North-Holland."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Ore, O. (1962). Theory of Graphs, American Mathematical Society Colloquium Publications.","DOI":"10.1090\/coll\/038"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"839","DOI":"10.1016\/j.disc.2012.11.031","article-title":"Independent domination in graphs: A survey and recent results","volume":"313","author":"Goddard","year":"2013","journal-title":"Discret. Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1016\/S0021-9800(70)80011-4","article-title":"Some covering concepts in graphs","volume":"8","author":"Plummer","year":"1970","journal-title":"J. Comb. Theory"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1080\/16073606.1993.9631737","article-title":"Well-covered graphs: A survey","volume":"16","author":"Plummer","year":"1993","journal-title":"Quaest. Math."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Majeed, A., and Rauf, I. (2020). Graph theory: A comprehensive survey about graph theory applications in computer science and social networks. Inventions, 5.","DOI":"10.3390\/inventions5010010"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"112","DOI":"10.1016\/j.disc.2005.10.006","article-title":"On weakly connected domination in graphs II","volume":"305","author":"Domke","year":"2005","journal-title":"Discret. Math."},{"key":"ref_10","first-page":"27","article-title":"Excellent trees","volume":"34","author":"Fricke","year":"2002","journal-title":"Bull. Inst. Comb. Appl."},{"key":"ref_11","first-page":"20","article-title":"Well-covered graphs","volume":"2","author":"Ravindra","year":"1977","journal-title":"J. Comb. Inf. Syst. Sci."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/8\/1411\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:39:03Z","timestamp":1760164743000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/8\/1411"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,8,2]]},"references-count":11,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2021,8]]}},"alternative-id":["sym13081411"],"URL":"https:\/\/doi.org\/10.3390\/sym13081411","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2021,8,2]]}}}