Characterization of divergence by order via non-cyclic subgraphs

Let $G$ be a graph, let $n$ be the fixed path-order parameter, and let $HL^k(G)$ denote the graph obtained after applying the $P_n$-line graph operator $k$ times. A connected subgraph $G'$ of $HL^k(G)$ contains a cycle $C_m$ when it has a subgraph isomorphic to $C_m$.

Divergence characterization conjecture. The graph $G$ has a sequence that diverges by order if and only if there exists a $k$ such that $HL^k(G)$ has a connected subgraph $G'$ where $G'$ has a subgraph isomorphic to $C_m$ ($m\geq n$) but $G'\not\cong C_m$.

The preceding theorem proves the sufficiency of the stated structural condition; the conjecture asserts that this condition is also necessary.