In this paper, we study the graph induced by the $\textit{2-swap}$ permutation on words with a fixed Parikh vector. A $2$-swap is defined as a pair of positions $s = (i, j)$ where the word $w$ induced by the swap $s$ on $v$ is $v[1] v[2] \dots v[i - 1] v[j] v[i+1] \dots v[j - 1] v[i] v[j + 1] \dots v[n]$. With these permutations, we define the $\textit{Configuration Graph}$, $G(P)$ defined over a given Parikh vector. Each vertex in $G(P)$ corresponds to a unique word with the Parikh vector $P$, with an edge between any pair of words $v$ and $w$ if there exists a swap $s$ such that $v \circ s = w$. We provide several key combinatorial properties of this graph, including the exact diameter of this graph, the clique number of the graph, and the relationships between subgraphs within this graph. Additionally, we show that for every vertex in the graph, there exists a Hamiltonian path starting at this vertex. Finally, we provide an algorithm enumerating these paths from a given input word of length $n$ with a delay of at most $O(\log n)$ between outputting edges, requiring $O(n \log n)$ preprocessing.

翻译：本文研究了具有固定Parikh向量的词在$\textit{2-交换}$置换下诱导的图。定义一个2-交换为一对位置$s = (i, j)$，其中通过交换$s$作用于词$v$得到的词$w$为$v[1] v[2] \dots v[i - 1] v[j] v[i+1] \dots v[j - 1] v[i] v[j + 1] \dots v[n]$。利用这些置换，我们定义了在给定Parikh向量上的$\textit{配置图}$ $G(P)$。$G(P)$中每个顶点对应一个具有Parikh向量$P$的唯一词，且若存在一个交换$s$使得$v \circ s = w$，则词$v$和$w$之间有一条边。我们提供了该图的若干关键组合性质，包括该图的精确直径、团数以及子图间的关系。此外，我们证明对于图中每个顶点，存在一条以该顶点为起点的哈密顿路径。最后，我们给出一个从给定输入词（长度为$n$）枚举这些路径的算法，其输出边的延迟不超过$O(\log n)$，预处理时间为$O(n \log n)$。