连续顶点删除下具有长伪相似链的图
Graphs with Long Pseudosimilarity Chains under Consecutive Vertex Deletions
AI总结：
该研究定义图的伪相似深度，通过双时钟和中国余数构造得到能在顶点删除中保持伪相似性的图，证明伪相似性可在渐近完整的顶点删除序列中持续存在。
AI中文摘要：
伪相似顶点是指属于不同自同构轨道，删除后会产生同构图的顶点。经典研究已探讨了这类顶点的存在性、群论起源及大量此类顶点的构造。我们提出一个不同的递归问题：在每次删除一个当前为伪相似的顶点时，该过程能持续多久？我们定义了图的伪相似深度，并构造了连通图，使得该过程可在除了亚线性数量的顶点外全部进行。双时钟构造在图的阶数上给出了平方根级的缺失，而带有多个循环时钟的中国余数构造则得到了一个无限的非对称图族，其中活跃链外仅剩下对数级数量的顶点。该机制通过打破一个长的隐藏自同构轨道并每次将断点扩大一个顶点来实现伪相似性。因此，即使主构造中遇到的每个图都是非对称的，伪相似性仍可在渐近完整的顶点删除序列中持续存在。
英文摘要：
Pseudosimilar vertices are vertices in distinct automorphism orbits whose deletions produce isomorphic graphs. Classical work has studied the existence, group-theoretic origin, and construction of large sets of such vertices. We ask a different recursive question: how long can one repeatedly delete a vertex that is pseudosimilar at the moment of deletion? We define the pseudosimilarity depth of a graph and construct connected graphs in which this process continues through all but a sublinear number of vertices. A two-clock construction gives a square-root deficit uniformly in the order, while a Chinese-remainder construction with many cyclic clocks yields an infinite family of asymmetric graphs with only a polylogarithmic number of vertices left outside the active chain. The mechanism realizes pseudosimilarity by breaking a long hidden automorphism orbit and enlarging the break one vertex at a time. Thus pseudosimilarity can persist through an asymptotically full sequence of vertex deletions, even though every graph encountered in the main construction is asymmetric.
