Unique minimal convergent subgraph conjecture

Let $\Lambda_n$ denote the class of graphs whose $P_n$-line graph sequences converge, and let $\lambda_n$ denote the class of minimally $n$-convergent graphs. For graphs $G$ and $G'$, write $G'\subseteq G$ when $G'$ is a subgraph of $G$.

Unique minimal subgraph conjecture. If $G\in\Lambda_n$ and $G$ is not the disconnected union of two graphs in $\Lambda_n$, then there exists a unique graph $G'\in\lambda_n$ such that $G'\subseteq G$.

The conjecture is proposed as a further direction in the study of minimally $n$-convergent graphs; the preceding theorem gives a related preservation result for unicyclic components under an additional girth hypothesis.