{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,25]],"date-time":"2026-01-25T03:34:01Z","timestamp":1769312041560,"version":"3.49.0"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Let $G$ be a connected $n$-vertex graph in a proper minor-closed class $\\mathcal G$. We prove that the extension complexity of the spanning tree polytope of $G$ is $O(n^{3\/2})$. This improves on the $O(n^2)$ bounds following from the work of Wong (1980) and Martin (1991). It also extends a result of Fiorini, Huynh, Joret, and Pashkovich (2017), who obtained a $O(n^{3\/2})$ bound for graphs embedded in a fixed surface. Our proof works more generally for all graph classes admitting strongly sublinear balanced separators: We prove that for every constant $\\beta$ with $0&lt;\\beta&lt;1$, if $\\mathcal G$ is a graph class closed under induced subgraphs such that all $n$-vertex graphs in $\\mathcal G$ have balanced separators of size $O(n^\\beta)$, then the extension complexity of the spanning tree polytope of every connected $n$-vertex graph in $\\mathcal{G}$ is $O(n^{1+\\beta})$. We in fact give two proofs of this result, one is a direct construction of the extended formulation, the other is via communication protocols. Using the latter approach we also give a short proof of the $O(n)$ bound for planar graphs due to Williams (2002).<\/jats:p>","DOI":"10.37236\/10522","type":"journal-article","created":{"date-parts":[[2021,12,17]],"date-time":"2021-12-17T01:48:42Z","timestamp":1639705722000},"source":"Crossref","is-referenced-by-count":4,"title":["Smaller Extended Formulations for Spanning Tree Polytopes in Minor-closed Classes and Beyond"],"prefix":"10.37236","volume":"28","author":[{"given":"Manuel","family":"Aprile","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Samuel","family":"Fiorini","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tony","family":"Huynh","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Gwena\u00ebl","family":"Joret","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"David R.","family":"Wood","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2021,12,17]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v28i4p47\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v28i4p47\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,12,17]],"date-time":"2021-12-17T01:48:42Z","timestamp":1639705722000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v28i4p47"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,12,17]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2021,10,8]]}},"URL":"https:\/\/doi.org\/10.37236\/10522","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,12,17]]},"article-number":"P4.47"}}