{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,21]],"date-time":"2026-01-21T14:46:10Z","timestamp":1769006770643,"version":"3.49.0"},"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>The number of vertices missed by a maximum matching in a graph $G$ is the multiplicity of zero as a root of the matchings polynomial $\\mu(G,x)$ of $G$, and hence many results in matching theory can be expressed in terms of this multiplicity. Thus, if $\\mathrm{mult}(\\theta,G)$ denotes the multiplicity of $\\theta$ as a zero of $\\mu(G,x)$, then Gallai's lemma is equivalent to the assertion that if $\\mathrm{mult}(\\theta,G\\setminus u) &lt; \\mathrm{mult}(\\theta,G)$ for each vertex $u$ of $G$, then $\\mathrm{mult}(\\theta,G)=1$. This paper extends a number of results in matching theory to results concerning $\\mathrm{mult}(\\theta,G)$, where $\\theta$ is not necessarily zero. If $P$ is a path in $G$ then $G\\setminus P$ denotes the graph got by deleting the vertices of $P$ from $G$. We prove that $\\mathrm{mult}(\\theta,G\\setminus P)\\ge\\mathrm{mult}(\\theta,G)-1$, and we say $P$ is $\\theta$-essential when equality holds. We show that if, all paths in $G$ are $\\theta$-essential, then $\\mathrm{mult}(\\theta,G)=1$. We define $G$ to be $\\theta$-critical if all vertices in $G$ are $\\theta$-essential and $\\mathrm{mult}(\\theta,G)=1$. We prove that if $\\mathrm{mult}(\\theta,G)=k$ then there is an induced subgraph $H$ with exactly $k$ $\\theta$-critical components, and the vertices in $G\\setminus H$ are covered by $k$ disjoint paths.<\/jats:p>","DOI":"10.37236\/1202","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T01:26:11Z","timestamp":1578705971000},"source":"Crossref","is-referenced-by-count":12,"title":["Algebraic Matching Theory"],"prefix":"10.37236","volume":"2","author":[{"given":"C. D.","family":"Godsil","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[1995,4,11]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v2i1r8\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v2i1r8\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T09:32:14Z","timestamp":1579339934000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v2i1r8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1995,4,11]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[1995,1,1]]}},"URL":"https:\/\/doi.org\/10.37236\/1202","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[1995,4,11]]},"article-number":"R8"}}