We investigate the Independent Set Reconfiguration problem under the Token Sliding rule. Let $I$ be an independent set of a simple undirected graph $G$. Suppose each vertex of $I$ has a token placed on it. The tokens can be moved one at a time by sliding along the edges of $G$, ensuring that after each move, the vertices with tokens always form an independent set of $G$.
The problem we address is whether we can transform $I$ into $I'$ through a sequence of steps, each involving substituting a vertex in the current independent set with one of its neighbors to obtain another independent set. This problem, known as determining if one independent set of a graph is