{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T15:41:03Z","timestamp":1753890063164,"version":"3.41.2"},"reference-count":0,"publisher":"Centre pour la Communication Scientifique Directe (CCSD)","issue":"Graph Theory","license":[{"start":{"date-parts":[[2018,11,20]],"date-time":"2018-11-20T00:00:00Z","timestamp":1542672000000},"content-version":"am","delay-in-days":0,"URL":"https:\/\/arxiv.org\/licenses\/nonexclusive-distrib\/1.0"},{"start":{"date-parts":[[2018,11,20]],"date-time":"2018-11-20T00:00:00Z","timestamp":1542672000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/arxiv.org\/licenses\/nonexclusive-distrib\/1.0"},{"start":{"date-parts":[[2018,11,20]],"date-time":"2018-11-20T00:00:00Z","timestamp":1542672000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/arxiv.org\/licenses\/nonexclusive-distrib\/1.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"accepted":{"date-parts":[[2025,3,31]]},"abstract":"<jats:p>We consider a relaxation of the concept of well-covered graphs, which are graphs with all maximal independent sets of the same size. The extent to which a graph fails to be well-covered can be measured by its independence gap, defined as the difference between the maximum and minimum sizes of a maximal independent set in $G$. While the well-covered graphs are exactly the graphs of independence gap zero, we investigate in this paper graphs of independence gap one, which we also call almost well-covered graphs. Previous works due to Finbow et al. (1994) and Barbosa et al. (2013) have implications for the structure of almost well-covered graphs of girth at least $k$ for $k\\in \\{7,8\\}$. We focus on almost well-covered graphs of girth at least $6$. We show that every graph in this class has at most two vertices each of which is adjacent to exactly $2$ leaves. We give efficiently testable characterizations of almost well-covered graphs of girth at least $6$ having exactly one or exactly two such vertices. Building on these results, we develop a polynomial-time recognition algorithm of almost well-covered $\\{C_3,C_4,C_5,C_7\\}$-free graphs.<\/jats:p>","DOI":"10.23638\/dmtcs-20-2-17","type":"journal-article","created":{"date-parts":[[2025,4,3]],"date-time":"2025-04-03T16:49:51Z","timestamp":1743698991000},"source":"Crossref","is-referenced-by-count":0,"title":["On Almost Well-Covered Graphs of Girth at Least 6"],"prefix":"10.23638","volume":"vol. 20 no. 2","author":[{"given":"T\u0131naz","family":"Ekim","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Didem","family":"G\u00f6z\u00fcpek","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ademir","family":"Hujdurovi\u0107","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Martin","family":"Milani\u010d","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"25203","published-online":{"date-parts":[[2018,11,20]]},"container-title":["Discrete Mathematics &amp; Theoretical Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/arxiv.org\/pdf\/1708.04632v4","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/arxiv.org\/pdf\/1708.04632v4","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,4,3]],"date-time":"2025-04-03T16:49:51Z","timestamp":1743698991000},"score":1,"resource":{"primary":{"URL":"http:\/\/dmtcs.episciences.org\/4514"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,11,20]]},"references-count":0,"journal-issue":{"issue":"Graph Theory","published-online":{"date-parts":[[2018,11,20]]}},"URL":"https:\/\/doi.org\/10.23638\/dmtcs-20-2-17","relation":{"has-preprint":[{"id-type":"arxiv","id":"1708.04632v3","asserted-by":"subject"},{"id-type":"arxiv","id":"1708.04632v2","asserted-by":"subject"}],"is-same-as":[{"id-type":"arxiv","id":"1708.04632","asserted-by":"subject"},{"id-type":"doi","id":"10.48550\/arXiv.1708.04632","asserted-by":"subject"}]},"ISSN":["1365-8050"],"issn-type":[{"type":"electronic","value":"1365-8050"}],"subject":[],"published":{"date-parts":[[2018,11,20]]},"article-number":"4514"}}