{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:19Z","timestamp":1753893859691,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>For a graph $G$, let $\\gamma(G)$ denote the domination number of $G$ and let $\\delta(G)$ denote the minimum degree among the vertices of $G$. A vertex $x$ is called a bad-cut-vertex of $G$ if $G-x$ contains a component, $C_x$, which is an induced $4$-cycle and $x$ is adjacent to at least one but at most three vertices on $C_x$. A cycle $C$ is called a special-cycle if $C$ is a $5$-cycle in $G$ such that if $u$ and $v$ are consecutive vertices on $C$, then at least one of $u$ and $v$ has degree $2$ in $G$. We let ${\\rm bc}(G)$ denote the number of bad-cut-vertices in $G$, and ${\\rm sc}(G)$ the maximum number of vertex disjoint special-cycles in $G$ that contain no bad-cut-vertices. We say that a graph is $(C_4,C_5)$-free if it has no induced $4$-cycle or $5$-cycle. Bruce Reed [Paths, stars and the number three. Combin.  Probab. Comput. 5 (1996), 277\u2013295] showed that if $G$ is a graph of order $n$ with $\\delta(G) \\ge 3$, then $\\gamma(G) \\le 3n\/8$. In this paper, we relax the minimum degree condition from three to two. Let $G$ be a connected graph of order $n \\ge 14$ with $\\delta(G) \\ge 2$.  As an application of Reed's result, we show that $\\gamma(G) \\le \\frac{1}{8} ( 3n + {\\rm sc}(G) + {\\rm bc}(G))$. As a consequence of this result, we have that (i) $\\gamma(G) \\le 2n\/5$; (ii) if $G$ contains no special-cycle and no bad-cut-vertex, then $\\gamma(G) \\le 3n\/8$; (iii) if $G$ is $(C_4,C_5)$-free, then $\\gamma(G) \\le 3n\/8$; (iv) if $G$ is $2$-connected and $d_G(u) + d_G(v) \\ge 5$ for every two adjacent vertices $u$ and $v$, then $\\gamma(G) \\le 3n\/8$. All bounds are sharp.<\/jats:p>","DOI":"10.37236\/499","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T03:57:13Z","timestamp":1578715033000},"source":"Crossref","is-referenced-by-count":4,"title":["A New Bound on the Domination Number of Graphs with Minimum Degree Two"],"prefix":"10.37236","volume":"18","author":[{"given":"Michael A.","family":"Henning","sequence":"first","affiliation":[]},{"given":"Ingo","family":"Schiermeyer","sequence":"additional","affiliation":[]},{"given":"Anders","family":"Yeo","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2011,1,5]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i1p12\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i1p12\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T23:18:06Z","timestamp":1579303086000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v18i1p12"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,1,5]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2011,1,5]]}},"URL":"https:\/\/doi.org\/10.37236\/499","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2011,1,5]]},"article-number":"P12"}}