SIAM Journal on Computing, Ahead of Print.
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 $\varepsilon > 0$). 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].

中文翻译：

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随机游走和禁止的未成年人I：有界度图上次要封闭属性的$ n ^ {1/2 + o（1）} $查询单面测试器

《 SIAM计算杂志》，预印本。
假设$ G $是具有$ n $个顶点的无向有界度图。修复一个有限的图$ H $，并假设必须从$ G $中除去$ \ varepsilon n $边使其变为$ H $ -minor-free（对于一些小的常数$ \ varepsilon> 0 $）。我们给出了$ n ^ {1/2 + o（1）} $时间的随机过程，该过程很有可能在这样的图中找到$ H $ -minor。作为一种应用，假设必须从有界度图$ G $中除去$ \ varepsilon n $边以使其平坦。此结果表明，在相同的运行时间下，算法会在$ G $中生成$ K_ {3,3} $-或$ K_5 $ -minor。没有先前的亚线性时间界限可解决此问题。通过图次要定理，我们得到任何次要封闭性质的相似结果。直到$ n ^ {o（1）} $个因子，这才解决了Benjamini，Schramm和Shapira的猜想。Math。，223（2010），pp。[2200--2218]中存在用于小封闭特性的单面特性测试器。此外，由于Czumaj等人的下限$ \ Omega（\ sqrt {n}）$，我们的算法几乎是最优的。[Random Structures Algorithms，45（2014），pp.139--184]。在进行这项工作之前，唯一以非平凡的单面性质测试员以$ H $-次要自由度而闻名的$ H $的图如下：$ H $是森林或循环[Czumaj等，Random Structures Algorithms，45（2014），pp.139--184]，$ K_ {2，k} $，$（k \ times 2）$-grid和$ k $ -circus [Fichtenberger et al。，preprint， arXiv：1707.06126v1，2017]。