识别正则镶嵌的子图
Recognizing Subgraphs of Regular Tilings
AI总结：
研究识别正则镶嵌图的子图问题，发现双曲镶嵌可高效解决，而欧几里得镶嵌则为NP难问题。
摘要：
对于p,q≥2，{p,q}镶嵌图是平面图T_{p,q}，其中所有面都是长度为p的循环，所有顶点都有度数q。我们给出了识别这些图的（诱导）子图问题的算法，如下。对于1/p+1/q>1/2，这些图对应于球面的正则镶嵌。这些图是有限的，因此可以以常数时间识别其（诱导）子图。对于1/p+1/q=1/2，这些图对应于欧几里得平面的正则镶嵌。对于欧几里得正方形网格T_{4,4}，Bhatt和Cosmadakis（IPL'87）证明识别子图是NP难的，即使输入图是树。我们证明一个简单的分治算法在所有欧几里得镶嵌中都能达到亚指数运行时间，并且通过已知的减少观察到在T_{4,4}中存在几乎匹配的下界。对于1/p+1/q<1/2，这些图对应于双曲平面的正则镶嵌。作为我们的主要贡献，我们证明决定一个n顶点图是否同构于镶嵌T_{p,q}的子图可以在准多项式（n^{O(log n)}）时间内完成，对于任何固定的q。我们的双曲情况结果表明，其复杂性显著低于欧几里得变种，并且不太可能是NP难的。欧几里得结果也表明，即使所讨论的图是树，该问题可能最大难度。因此，已知的双曲镶嵌子图的树宽界限本身并不导致高效的算法。相反，我们使用镶嵌图内的凸包，其在双曲镶嵌中具有若干有利的属性。我们的关键技术洞察是，平面子图同构可以通过动态规划计算，该动态规划构建了解子图凸包的球面切割分解。
英文摘要：
For $p,q\ge2$ the $\{p,q\}$-tiling graph is the (finite or infinite) planar graph $T_{p,q}$ where all faces are cycles of length $p$ and all vertices have degree $q$. We give algorithms for the problem of recognizing (induced) subgraphs of these graphs, as follows. - For $1/p+1/q>1/2$, these graphs correspond to regular tilings of the sphere. These graphs are finite, thus recognizing their (induced) subgraphs can be done in constant time. - For $1/p+1/q=1/2$, these graphs correspond to regular tilings of the Euclidean plane. For the Euclidean square grid $T_{4,4}$ Bhatt and Cosmadakis (IPL'87) showed that recognizing subgraphs is NP-hard, even if the input graph is a tree. We show that a simple divide-and conquer algorithm achieves a subexponential running time in all Euclidean tilings, and we observe that there is an almost matching lower bound in $T_{4,4}$ under the Exponential Time Hypothesis via known reductions. - For $1/p+1/q<1/2$, these graphs correspond to regular tilings of the hyperbolic plane. As our main contribution, we show that deciding if an $n$-vertex graph is isomorphic to a subgraph of the tiling $T_{p,q}$ can be done in quasi-polynomial ($n^{O(\log n)}$) time for any fixed $q$. Our results for the hyperbolic case show that it has significantly lower complexity than the Euclidean variant, and it is unlikely to be NP-hard. The Euclidean results also suggest that the problem can be maximally hard even if the graph in question is a tree. Consequently, the known treewidth bounds for subgraphs of hyperbolic tilings do not lead to an efficient algorithm by themselves. Instead, we use convex hulls within the tiling graph, which have several desirable properties in hyperbolic tilings. Our key technical insight is that planar subgraph isomorphism can be computed via a dynamic program that builds a sphere cut decomposition of a solution subgraph's convex hull.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
