书籍、走廊与社交蝴蝶：关于滑动拼图的一个注记
Books, Hallways and Social Butterflies: A Note on Sliding Block Puzzles
AI总结：
本文推广Wilson对滑动拼图连通性的刻画至任意块数，确定连通所需空顶点数，并给出判断任意两构型是否连通的算法。
摘要：
回顾经典的15拼图，它由4×4网格中的15个滑动块组成。众所周知，该拼图的构型空间由两个连通分量组成，分别对应对称群$S_{15}$的奇置换和偶置换。1974年，Wilson将滑动拼图从4×4网格推广到任意图（考虑在具有$n$个顶点的图上有$n-1$个滑动块），并刻画了相应构型空间连通时的图。在本工作中，我们将Wilson的刻画推广到具有任意数量滑动块的滑动拼图（可能留下不止一个空顶点）。对于任意图，我们确定需要多少个空顶点才能使相应的构型空间连通，更一般地，我们提供了一个算法来确定任意两个构型是否连通。我们的工作也可以在“朋友与陌生人图”的框架下解释，其中空顶点对应“社交蝴蝶”，滑动块对应“不合群”的人。
英文摘要：
Recall the classical 15-puzzle, consisting of 15 sliding blocks in a $4\times 4$ grid. Famously, the configuration space of this puzzle consists of two connected components, corresponding to the odd and even permutations of the symmetric group $S_{15}$. In 1974, Wilson generalised sliding block puzzles beyond the $4\times 4$ grid to arbitrary graphs (considering $n-1$ sliding blocks on a graph with $n$ vertices), and characterised the graphs for which the corresponding configuration space is connected. In this work, we extend Wilson's characterisation to sliding block puzzles with an arbitrary number of blocks (potentially leaving more than one empty vertex). For any graph, we determine how many empty vertices are necessary to connect the corresponding configuration space, and more generally we provide an algorithm to determine whether any two configurations are connected. Our work may also be interpreted within the framework of "Friends and Strangers graphs", where empty vertices correspond to "social butterflies" and sliding blocks to "asocial" people.
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。
