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：本文研究由固定Parikh向量下词上的$\textit{2-交换}$置换所诱导的图。2-交换定义为位置对$s = (i, j)$，其中交换$s$作用于$v$得到的词$w$为$v[1] v[2] \dots v[i - 1] v[j] v[i+1] \dots v[j - 1] v[i] v[j + 1] \dots v[n]$。基于此类置换，我们定义了给定Parikh向量上的$\textit{构型图}$ $G(P)$。$G(P)$中每个顶点对应一个具有Parikh向量$P$的唯一词，若存在交换$s$使得$v \circ s = w$，则任意一对词$v$和$w$之间存在边。我们给出了该图的若干关键组合性质，包括图的精确直径、图的团数，以及图中子图间的关系。此外，我们证明图中每个顶点均存在一条从该顶点出发的哈密顿路径。最后，我们提出一个算法，可在给定长度为$n$的输入词后枚举这些路径，其输出边之间的延迟至多为$O(\log n)$，且需要$O(n \log n)$的预处理时间。