{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T19:28:18Z","timestamp":1787340498997,"version":"build-2736575974"},"reference-count":37,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"6","funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["CCF-1319080"],"award-info":[{"award-number":["CCF-1319080"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["CCF-1740850"],"award-info":[{"award-number":["CCF-1740850"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Comput."],"published-print":{"date-parts":[[2023,12]]},"abstract":"<jats:p>Let $G$ be an undirected, bounded degree graph with $n$ vertices. Fix a finite graph $H$, and suppose one must remove $\\varepsilon n$ edges from $G$ to make it $H$-minor-free (for some small constant $\\varepsilon &gt; 0$). We give an $n^{1\/2+o(1)}$-time randomized procedure that, with high probability, finds an $H$-minor in such a graph. As an application, suppose one must remove $\\varepsilon n$ edges from a bounded degree graph $G$ to make it planar. This result implies an algorithm, with the same running time, that produces a $K_{3,3}$- or $K_5$-minor in $G$. No prior sublinear time bound was known for this problem. By the graph minor theorem, we get an analogous result for any minor-closed property. Up to $n^{o(1)}$ factors, this resolves a conjecture of Benjamini, Schramm, and Shapira [ Adv. Math., 223 (2010), pp. 2200--2218] on the existence of one-sided property testers for minor-closed properties. Furthermore, our algorithm is nearly optimal by an $\\Omega(\\sqrt{n})$ lower bound of Czumaj et al. [ Random Structures Algorithms, 45 (2014), pp. 139--184]. Prior to this work, the only graphs $H$ for which nontrivial one-sided property testers were known for $H$-minor-freeness were the following: $H$ being a forest or a cycle [Czumaj et al., Random Structures Algorithms, 45 (2014), pp. 139--184], $K_{2,k}$, $(k\\times 2)$-grid, and the $k$-circus [Fichtenberger et al., preprint, arXiv:1707.06126v1, 2017].<\/jats:p>","DOI":"10.1137\/19m1245463","type":"journal-article","created":{"date-parts":[[2020,12,18]],"date-time":"2020-12-18T16:02:47Z","timestamp":1608307367000},"page":"FOCS18-216-FOCS18-249","source":"Crossref","is-referenced-by-count":0,"title":["Random Walks and Forbidden Minors I: An $n^{1\/2+o(1)}$-Query One-Sided Tester for Minor Closed Properties on Bounded Degree Graphs"],"prefix":"10.1137","volume":"52","author":[{"given":"Akash","family":"Kumar","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"C.","family":"Seshadhri","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Andrew","family":"Stolman","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2020,12,18]]},"reference":[{"key":"atypb1","volume-title":"Reversible Markov Chains and Random Walks on Graphs, unfinished monograph","author":"Aldous D.","year":"2002"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1137\/060667177"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1109\/FOCS.2006.44"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1145\/2775105"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1016\/j.aim.2009.10.018"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1098-2418(1999010)14:1<87::AID-RSA5>3.0.CO;2-O"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20462"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1145\/2746539.2746618"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1137\/070681831"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1017\/S096354831000012X"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-14279-6"},{"key":"atypb12","first-page":"530","volume-title":"RANDOM 2011","author":"Edelman A.","year":"2011"},{"key":"atypb13","volume-title":"On Testing Minor-Freeness in Bounded Degree Graphs with One-Sided Error, preprint, https:\/\/arxiv.org\/abs\/1707.06126v1","author":"Fichtenberger H.","year":"2017"},{"key":"atypb14","volume-title":"Proceedings of the 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018","author":"Fichtenberger H.","year":"2018"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1109\/FOCS.2012.85"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1017\/9781108135252"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1007\/s004930050060"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1007\/s00453-001-0078-7"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-22670-0_9"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1109\/FOCS.2009.77"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1145\/321850.321852"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1137\/110837863"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1137\/100802980"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1016\/j.jctb.2011.07.004"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1109\/SFCS.1996.548467"},{"key":"atypb26","volume-title":"Finding Forbidden Minors in Sublinear Time: A $o(n^{1\/2 + o(1)})$-Query One-Sided Tester for Minor Closed Properties on Bounded Degree Graphs, preprint, https:\/\/arxiv.org\/abs\/1805.08187v1","author":"Kumar A.","year":"2018"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.4064\/fm-15-1-271-283"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1145\/2629508"},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.1090\/S0273-0979-05-01088-8"},{"key":"atypb30","doi-asserted-by":"publisher","DOI":"10.1109\/FSCS.1990.89553"},{"key":"atypb31","doi-asserted-by":"publisher","DOI":"10.1016\/j.ic.2009.09.002"},{"key":"atypb32","doi-asserted-by":"publisher","DOI":"10.1016\/j.jctb.2004.08.001"},{"key":"atypb33","unstructured":"D. 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