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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 apair of positions $s = (i, j)$ where the word $w$ induced by the swap $s$ on$v$ is $v[1] v[2] \dots v[i - 1] v[j] v[i+1] \dots v[j - 1] v[i] v[j + 1] \dotsv[n]$. With these permutations, we define the $\textit{Configuration Graph}$,$G(P)$ defined over a given Parikh vector. Each vertex in $G(P)$ corresponds toa unique word with the Parikh vector $P$, with an edge between any pair ofwords $v$ and $w$ if there exists a swap $s$ such that $v \circ s = w$. Weprovide several key combinatorial properties of this graph, including the exactdiameter of this graph, the clique number of the graph, and the relationshipsbetween subgraphs within this graph. Additionally, we show that for everyvertex in the graph, there exists a Hamiltonian path starting at this vertex.Finally, we provide an algorithm enumerating these paths from a given inputword of length $n$ with a delay of at most $O(\log n)$ between outputtingedges, requiring $O(n \log n)$ preprocessing.

Structural and Combinatorial Properties of 2-swap Word Permutation Graphs