{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:43:17Z","timestamp":1760060597291,"version":"build-2065373602"},"reference-count":26,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2025,9,5]],"date-time":"2025-09-05T00:00:00Z","timestamp":1757030400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Huizhou City Basic Education educational science research project","award":["2023hzkt193","158 [2023]","12171402","2023hzkt193"],"award-info":[{"award-number":["2023hzkt193","158 [2023]","12171402","2023hzkt193"]}]},{"name":"Teaching Quality and Teaching Reform Project of Huizhou University","award":["2023hzkt193","158 [2023]","12171402","2023hzkt193"],"award-info":[{"award-number":["2023hzkt193","158 [2023]","12171402","2023hzkt193"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["2023hzkt193","158 [2023]","12171402","2023hzkt193"],"award-info":[{"award-number":["2023hzkt193","158 [2023]","12171402","2023hzkt193"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Huizhou City Basic Education educational science research project","award":["2023hzkt193","158 [2023]","12171402","2023hzkt193"],"award-info":[{"award-number":["2023hzkt193","158 [2023]","12171402","2023hzkt193"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>A single circular DNA or polypeptide strand can be used to construct nanostructures based on the mathematical concept of strong traces. It is known that if a special face graph of a 2-connected plane graph G is a tree, then G has a thickened graph F(G) with only one boundary component. However, to ensure that this thickened graph contains only a single boundary component, it is not necessary for the special face graph of G to be a tree. In this paper, we summarize a combinatorial method, called the flat-based operations, for obtaining a matrix M0 where det(M0) determines whether F(G) associated with G contains just one boundary component. By applying this method, we can easily determine if the number of boundary components of some F(G) is one. Further, we characterize a type of 2-connected plane graph whose thickened graph has more than one boundary component and then has no antiparallel strong trace.<\/jats:p>","DOI":"10.3390\/sym17091463","type":"journal-article","created":{"date-parts":[[2025,9,5]],"date-time":"2025-09-05T09:51:21Z","timestamp":1757065881000},"page":"1463","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Combinatorial Method for the Number of Components of DNA and Polypeptide Cages"],"prefix":"10.3390","volume":"17","author":[{"given":"Jiajun","family":"Ning","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, Xiamen University, Xiamen 361005, China"}]},{"given":"Xiao-Sheng","family":"Cheng","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Huizhou University, Huizhou 516007, China"}]}],"member":"1968","published-online":{"date-parts":[[2025,9,5]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1795","DOI":"10.1126\/science.1154533","article-title":"Assembling materials with DNA as the guide","volume":"321","author":"Aldaye","year":"2008","journal-title":"Science"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1146\/annurev-biochem-060308-102244","article-title":"Nanomaterials based on DNA","volume":"79","author":"Seeman","year":"2010","journal-title":"Annu. 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