Isomorphism Testing for Graphs Excluding Small Topological Subgraphs
Daniel Neuen
备注：
42 pages, 3 figures, full version of a paper accepted at SODA 2022; second and third version improve the presentation of the results. arXiv admin note: text overlap with arXiv:2004.07671
我们提出一种同构测试，在排除某个h顶点图作为拓扑子图的所有n顶点图上，其运行时间为$n^{\text{polylog}(h)}$。此前结果表明，此类图的同构测试可在$n^{\text{polylog}(n)}$时间内完成（Babai，STOC 2016），或在$n^{f(h)}$时间内完成（其中f为某函数，Grohe和Marx，SIAM J. Comp.，2015）。我们的结果还统一并扩展了此前针对最大度为d的图（运行时间$n^{\text{polylog}(d)}$，SIAM J. Comp.，2023）和Hadwiger数为h的图（运行时间$n^{\text{polylog}(h)}$，SIAM J. Comp.，2023）的同构测试。
英文摘要：
We give an isomorphism test that runs in time $n^{\operatorname{polylog}(h)}$ on all $n$-vertex graphs excluding some $h$-vertex vertex graph as a topological subgraph. Previous results state that isomorphism for such graphs can be tested in time $n^{\operatorname{polylog}(n)}$ (Babai, STOC 2016) and $n^{f(h)}$ for some function $f$ (Grohe and Marx, SIAM J. Comp., 2015).
Our result also unifies and extends previous isomorphism tests for graphs of maximum degree $d$ running in time $n^{\operatorname{polylog}(d)}$ (SIAM J. Comp., 2023) and for graphs of Hadwiger number $h$ running in time $n^{\operatorname{polylog}(h)}$ (SIAM J. Comp., 2023).
We generalize the structure theorem of Robertson and Seymour for graphs excluding a fixed graph $H$ as a minor to graphs excluding $H$ as a topological subgraph. We prove that for a fixed $H$, every graph excluding $H$ as a topological subgraph has a tree decomposition where each part is either "almost embeddable" to a fixed surface or has bounded degree with the exception of a bounded number of vertices. Furthermore, we prove that such a decomposition is computable by an algorithm that is fixed-parameter tractable with parameter $|H|$. We present two algorithmic applications of our structure …
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