Mynhardt–Roux path and cycle non-realisation conjecture for irredundance graphs

For a graph $G$, an $\operatorname{IR}$-graph is a graph realised as the slide graph of the irredundance sets of $G$. Let $P_n$ and $C_n$ denote the path and cycle on $n$ vertices.

Mynhardt–Roux's conjecture. For every $n\geq3$, $P_n$ is not an $\operatorname{IR}$-graph, and for every $n\geq5$, $C_n$ is not an $\operatorname{IR}$-graph.

This conjecture concerns which graphs can arise from slide reconfiguration of irredundance sets. The cited work had already established several non-realisation results, including stars, certain short paths and cycles, and some smallest irredundance trees; the assertions for all paths and cycles in the stated ranges remain open in the supplied context.