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I am going through the paper https://pubsonline.informs.org/doi/abs/10.1287/moor.10.4.701 which is A Note on Finding Minimum-Cost Edge-Disjoint Spanning Trees and the authors are James Roskind and Robert E. Tarjan. In it, there is an algorithm to generate an augmenting sequence used in updating edge-disjoint forests.

If $F$ is a forest and $e = \{v, w\}$ is an edge such that $v$ and $w$ are in the same tree of $F$, we define $F(e)$ to be the unique path in $F$ joining $v$ and $w$. If $i$ is any integer, we define $i+ = (i $ mod $k) + 1$. For a given set of edge-disjoint forests $F_1$, $F_2$, $\ldots$, $F_k$ and edges $e_0$ and $e_l$, a swap sequence from $e_0$ to $e_l$ is a sequence of edges $e_0$, $e_1$, . . , $e_l$ such that, for $j \in [0 \ldots l- 1]$, $e_{j+1} \in F_{j+} (e_j)$. The swap sequence is augmenting if $e_0$ is in none of the forests $F_1$, $F_2$, $\ldots$, $F_k$, the endpoints of $e_l$, are in different trees of $F_{l+}$, and the swap sequence is minimal in the sense that there are no two edges $e_{j'}$ and $e_j$, such that $j' > j + 1$ and $e_j' \in F_{j+}$

The definition for an augmenting swap sequence seems very arbitrary to me at the moment, as I am unable to visualize it properly. I do understand what it is saying, but I am not understanding how this definition is well-motivated. An example would be very great in clarifying this

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