Let $G$ be any graph and $f : V \rightarrow V$ describe an initial configuration of tokens in the vertices. The only operation permitted is the adjacent swap $(u,v)$, a swap that exchange two tokens if and only if there is an edge between $u$ and $v$. A cycle decomposition of a graph is a way to represent the permutation by using directed cycles (self-cycles are "correct" vertices and are called fixed-points).
Suppose that individual cycle $C$ (for any $C$) is a instance of the problem where the graph $G$ continues the same, but the permutation is modified to only have the cycle $C$ and fixed-point cycles in every other vertice ($V(G) \setminus V(C)$). In this instance, it can be proved that the minimum number of swaps possible is $n-1$, such that $|C| = n$, to transform the configuration $f$ to the identity configuration that maps every node to itself (only fixed-point cycles)?
I suppose that something can be argued on the point that a swap can "fix" at most one token, unless is a swapon a cycle of size two. But I am having problems in structuring this. Does a simple invariant work here?