The nonexistence conjecture for nontrivial uniquely cycle-saturated graphs

Let $C_t$ denote the cycle on $t$ vertices. A graph is uniquely $C_t$-saturated if it contains no copy of $C_t$, but adding any edge from its complement creates exactly one copy of $C_t$. Such a graph is nontrivial if it has at least $t$ vertices.

Nonexistence conjecture. For $t\ge 6$, there are no nontrivial uniquely $C_t$-saturated graphs.

The paper proves that there are no nontrivial uniquely $C_6$-saturated or uniquely $C_7$-saturated graphs, and that for every $t\ge 6$ only finitely many uniquely $C_t$-saturated graphs exist. The conjecture asserts that this finite collection is empty for every $t\ge 6$.