{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,7]],"date-time":"2025-11-07T09:53:38Z","timestamp":1762509218395,"version":"3.41.2"},"reference-count":0,"publisher":"Centre pour la Communication Scientifique Directe (CCSD)","issue":"Graph Theory","license":[{"start":{"date-parts":[[2017,10,17]],"date-time":"2017-10-17T00:00:00Z","timestamp":1508198400000},"content-version":"am","delay-in-days":0,"URL":"https:\/\/arxiv.org\/licenses\/nonexclusive-distrib\/1.0"},{"start":{"date-parts":[[2017,10,17]],"date-time":"2017-10-17T00:00:00Z","timestamp":1508198400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/arxiv.org\/licenses\/nonexclusive-distrib\/1.0"},{"start":{"date-parts":[[2017,10,17]],"date-time":"2017-10-17T00:00:00Z","timestamp":1508198400000},"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>Let $G$ be a simple graph with a perfect matching. Deng and Zhang showed that the maximum anti-forcing number of $G$ is no more than the cyclomatic number. In this paper, we get a novel upper bound on the maximum anti-forcing number of $G$ and investigate the extremal graphs. If $G$ has a perfect matching $M$ whose anti-forcing number attains this upper bound, then we say $G$ is an extremal graph and $M$ is a nice perfect matching. We obtain an equivalent condition for the nice perfect matchings of $G$ and establish a one-to-one correspondence between the nice perfect matchings and the edge-involutions of $G$, which are the automorphisms $\\alpha$ of order two such that $v$ and $\\alpha(v)$ are adjacent for every vertex $v$. We demonstrate that all extremal graphs can be constructed from $K_2$ by implementing two expansion operations, and $G$ is extremal if and only if one factor in a Cartesian decomposition of $G$ is extremal. As examples, we have that all perfect matchings of the complete graph $K_{2n}$ and the complete bipartite graph $K_{n, n}$ are nice. Also we show that the hypercube $Q_n$, the folded hypercube $FQ_n$ ($n\\geq4$) and the enhanced hypercube $Q_{n, k}$ ($0\\leq k\\leq n-4$) have exactly $n$, $n+1$ and $n+1$ nice perfect matchings respectively.<\/jats:p><jats:p>Comment: 15 pages, 7 figures<\/jats:p>","DOI":"10.23638\/dmtcs-19-3-9","type":"journal-article","created":{"date-parts":[[2025,4,3]],"date-time":"2025-04-03T16:41:17Z","timestamp":1743698477000},"source":"Crossref","is-referenced-by-count":1,"title":["Tight upper bound on the maximum anti-forcing numbers of graphs"],"prefix":"10.23638","volume":"Vol. 19 no. 3","author":[{"given":"Lingjuan","family":"Shi","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5385-6687","authenticated-orcid":false,"given":"Heping","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"25203","published-online":{"date-parts":[[2017,10,17]]},"container-title":["Discrete Mathematics &amp; Theoretical Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/arxiv.org\/pdf\/1704.04124v3","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/arxiv.org\/pdf\/1704.04124v3","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,4,3]],"date-time":"2025-04-03T16:41:17Z","timestamp":1743698477000},"score":1,"resource":{"primary":{"URL":"http:\/\/dmtcs.episciences.org\/3267"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,10,17]]},"references-count":0,"journal-issue":{"issue":"Graph Theory","published-online":{"date-parts":[[2017,10,17]]}},"URL":"https:\/\/doi.org\/10.23638\/dmtcs-19-3-9","relation":{"has-preprint":[{"id-type":"arxiv","id":"1704.04124v2","asserted-by":"subject"},{"id-type":"arxiv","id":"1704.04124v1","asserted-by":"subject"}],"is-same-as":[{"id-type":"arxiv","id":"1704.04124","asserted-by":"subject"},{"id-type":"doi","id":"10.48550\/arXiv.1704.04124","asserted-by":"subject"}]},"ISSN":["1365-8050"],"issn-type":[{"type":"electronic","value":"1365-8050"}],"subject":[],"published":{"date-parts":[[2017,10,17]]},"article-number":"3267"}}