{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:42:08Z","timestamp":1760146928490,"version":"build-2065373602"},"reference-count":11,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2024,12,27]],"date-time":"2024-12-27T00:00:00Z","timestamp":1735257600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["11901268","12131013"],"award-info":[{"award-number":["11901268","12131013"]}]},{"name":"Fundamental Research Funds for the Universities of Liaoning Province (2024)","award":["11901268","12131013"],"award-info":[{"award-number":["11901268","12131013"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Let G be a graph and S\u2286V(G). If S is a dominating set of G, and for each vertex u\u2208V(G)\u2212S, there is a neighbor of u in S, denoted by v, such that (S\u2216{v})\u222a{u} is a dominating set of G, then S is a secure dominating set (SDS) of G. Conversely, S is a co-secure dominating set (CSDS) of G if S is a dominating set of G and for each vertex v in S,\u00a0V(G)\u2212S contains a neighbor of v, denoted by u, such that (S\u2216{v})\u222a{u} is a dominating set of G. The minimum cardinality of a CSDS (resp. SDS) of G is the co-secure (resp. secure) domination number of G. We use \u03b3cs(G) and \u03b3s(G) to denote the co-secure domination number and secure domination number of G, respectively. Arumugam et al. proposed two questions: (1) Characterize a graph G with \u03b3cs(G)=\u03b1(G), where \u03b1(G) is the independence number of G; (2) Characterize a graph G with \u03b3cs(G)=\u03b3s(G). In this paper, we characterize some forbidden induced subgraphs for a graph G with \u03b3cs(G)=\u03b1(G); moreover, we obtain that \u03b3cs(G)=\u03b3s(G) for each {K3,C5,P5}-free graph G with \u03b4(G)\u22652. Our conclusions can generalize some known results.<\/jats:p>","DOI":"10.3390\/axioms14010010","type":"journal-article","created":{"date-parts":[[2024,12,27]],"date-time":"2024-12-27T09:13:32Z","timestamp":1735290812000},"page":"10","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Co-Secure Domination Number in Some Graphs"],"prefix":"10.3390","volume":"14","author":[{"given":"Jiatong","family":"Cui","sequence":"first","affiliation":[{"name":"School of Mathematics, Liaoning Normal University, Dalian 116029, China"}]},{"given":"Tianhao","family":"Li","sequence":"additional","affiliation":[{"name":"School of Mathematics, Liaoning Normal University, Dalian 116029, China"}]},{"given":"Jiayuan","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Mathematics, Liaoning Normal University, Dalian 116029, China"}]},{"given":"Xiaodong","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Mathematics, Liaoning Normal University, Dalian 116029, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3091-3252","authenticated-orcid":false,"given":"Liming","family":"Xiong","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Beijing Key Laboratory on MCAACI, Beijing Institute of Technology, Beijing 100081, China"}]}],"member":"1968","published-online":{"date-parts":[[2024,12,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Bondy, J., and Murty, U. (1976). Graph Theory with Applications, American Elsevier.","DOI":"10.1007\/978-1-349-03521-2"},{"key":"ref_2","first-page":"19","article-title":"Protection of a graph","volume":"67","author":"Cockayne","year":"2005","journal-title":"Util. Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1016\/j.dam.2017.10.020","article-title":"On the secure domination numbers of maximal outerplanar graphs","volume":"236","author":"Araki","year":"2018","journal-title":"Discrete Appl. Math."},{"key":"ref_4","first-page":"87","article-title":"Secure domination, weak Roman domination and forbidden subgraphs","volume":"39","author":"Cockayne","year":"2003","journal-title":"Bull. Inst. Comb. Appl."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1016\/j.dam.2023.03.016","article-title":"A note on secure domination in C5-free graphs","volume":"333","author":"Degawa","year":"2023","journal-title":"Discrete Appl. Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1016\/j.dam.2021.11.008","article-title":"The secure domination number of Cartesian products of small graphs with paths and cycles","volume":"309","author":"Haythorpe","year":"2022","journal-title":"Discrete Appl. Math."},{"key":"ref_7","first-page":"289","article-title":"Co-secure Domination in Mycielski Graph","volume":"113","author":"Manjusha","year":"2020","journal-title":"J. Comb. Math. Comb. Comput."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"786","DOI":"10.1016\/j.ipl.2015.05.006","article-title":"On secure domination in graphs","volume":"115","author":"Merouane","year":"2015","journal-title":"Inform. Process. Lett."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Pothuvath, M., Iyer, R.R., Asiri, A., and Somasundaram, K. (2024). Co-Secure Domination in Jump Graphs for Enhanced Security. Mathematics, 12.","DOI":"10.3390\/math12193077"},{"key":"ref_10","first-page":"167","article-title":"Co-secure and secure domination in graphs","volume":"94","author":"Arumugan","year":"2014","journal-title":"Util. Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"158","DOI":"10.14445\/22315373\/IJMTT-V55P520","article-title":"Bounds on co-secure domination in graphs","volume":"55","author":"Joseph","year":"2018","journal-title":"Int. J. Math. Trends Technol."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/1\/10\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T17:01:23Z","timestamp":1760115683000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/1\/10"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,12,27]]},"references-count":11,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,1]]}},"alternative-id":["axioms14010010"],"URL":"https:\/\/doi.org\/10.3390\/axioms14010010","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2024,12,27]]}}}