{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:38Z","timestamp":1753893818206,"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>Let the matching polynomial of a graph $G$ be denoted by $\\mu (G,x)$. A graph $G$ is said to be $\\theta$-super positive if\u00a0 $\\mu(G,\\theta)\\neq 0$ and $\\mu(G\\setminus v,\\theta)=0$ for all $v\\in V(G)$. In particular, $G$ is $0$-super positive if and only if $G$ has a perfect matching. While much is known about $0$-super positive graphs, almost nothing is known about $\\theta$-super positive graphs for $\\theta \\neq 0$. This motivates us to investigate the structure of $\\theta$-super positive graphs in this paper. Though a $0$-super positive graph need not contain any cycle, we show that a $\\theta$-super positive graph with $\\theta \\neq 0$ must contain a cycle. We introduce two important types of $\\theta$-super positive graphs, namely $\\theta$-elementary and $\\theta$-base graphs. One of our main results is that any $\\theta$-super positive graph $G$ can be constructed by adding certain type of edges to a disjoint union of $\\theta$-base graphs; moreover, these $\\theta$-base graphs are uniquely determined by $G$. We also give a characterization of $\\theta$-elementary graphs: a graph $G$ is $\\theta$-elementary if and only if the set of all its $\\theta$-barrier sets form a partition of $V(G)$. Here, $\\theta$-elementary graphs and $\\theta$-barrier sets can be regarded as $\\theta$-analogue of elementary graphs and Tutte sets in classical matching theory.<\/jats:p>","DOI":"10.37236\/2041","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T03:32:29Z","timestamp":1578713549000},"source":"Crossref","is-referenced-by-count":3,"title":["Properties of $\\theta$-super positive graphs"],"prefix":"10.37236","volume":"19","author":[{"given":"Cheng Yeaw","family":"Ku","sequence":"first","affiliation":[]},{"given":"Kok Bin","family":"Wong","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2012,2,7]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v19i1p37\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v19i1p37\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T22:42:23Z","timestamp":1579300943000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v19i1p37"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,2,7]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2012,2,15]]}},"URL":"https:\/\/doi.org\/10.37236\/2041","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2012,2,7]]},"article-number":"P37"}}