Tuite–Thomas–Chartrand circulant-graph conjecture

Let $n$ be the order of a graph, and let a circulant graph be a graph whose vertices and adjacency relation are invariant under cyclic translation. Let $\operatorname{mp}(G)$ denote the monophonic position number and $\operatorname{diam}(G)$ the diameter. Tuite–Thomas–Chartrand's conjecture. For any $n\geq 11$, there is a circulant graph with order $n$, monophonic position number two, and diameter two.

The construction preceding the conjecture preserves these properties while changing the order; the existence of a circulant example for every $n\geq 11$ remains open.