编辑距离函数的累积点
Accumulation points of the edit distance function
摘要：
给定一个图的遗传性质$\mathcal H$和一些$p\in[0,1]$，编辑距离函数$\operatorname{ed}_{\mathcal H}(p)$是（渐近地）将密度为$p$的任何图转换为$\mathcal H$的成员所需的“编辑”（边添加加边删除）的最大比例。对于任何固定的$p\in[0,1]$，可以从称为着色正则图（CRG）的对象计算$\operatorname{ed}_{\mathcal H}(p)$。本文研究的是那些点$p\in[0,1]$，对于这些点，需要无穷多个CRG来计算任何包含$p$的开区间上的$\operatorname{ed}_{\mathcal H}$;这样的$p$称为聚点。我们表明，正如预期的那样，$p=0$和$p=1$确实是一些遗传特性的积累点，我们还确定了$\operatorname{ed}_{\mathcal H}$在这两个极值点的斜率。出乎意料的是，我们构造了一个聚点在$p=1/4$的遗传性质。最后，我们推导出一个显着的结构性质，这些CRG发生在聚集点。
英文摘要：
Given a hereditary property $\mathcal H$ of graphs and some $p\in[0,1]$, the edit distance function $\operatorname{ed}_{\mathcal H}(p)$ is (asymptotically) the maximum proportion of "edits" (edge-additions plus edge-deletions) necessary to transform any graph of density $p$ into a member of $\mathcal H$. For any fixed $p\in[0,1]$, $\operatorname{ed}_{\mathcal H}(p)$ can be computed from an object known as a colored regularity graph (CRG). This paper is concerned with those points $p\in[0,1]$ for which infinitely many CRGs are required to compute $\operatorname{ed}_{\mathcal H}$ on any open interval containing $p$; such a $p$ is called an accumulation point. We show that, as expected, $p=0$ and $p=1$ are indeed accumulation points for some hereditary properties; we additionally determine the slope of $\operatorname{ed}_{\mathcal H}$ at these two extreme points. Unexpectedly, we construct a hereditary property with an accumulation point at $p=1/4$. Finally, we derive a significant structural property about those CRGs which occur at accumulation points.
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。
