Associated-prime extension conjecture for separated stars on edge ideals

Let $G$ be a graph, let $I=I(G)$ be its edge ideal in a polynomial ring $R$ over a field, and let $f_1,\ldots,f_q$ be separated stars on $I$. For an ideal $J$, write $\operatorname{Ass}(R/J)$ for its associated primes and $\operatorname{Min}(R/J)$ for its minimal primes. The stars are linear forms as defined in the paper's separation framework.

Associated-prime extension conjecture. Every associated prime $P$ of $R/(I,f_1,\ldots,f_q)$ is of the form

$$P=(Q,f_1,\ldots,f_q)$$

for some $Q\in\operatorname{Ass}(R/I)=\operatorname{Min}(R/I)$; equivalently,

$$P\in\operatorname{Ass}(R/(I,f_1,\ldots,f_q))\quad\Longleftrightarrow\quad P=(Q,f_1,\ldots,f_q)\text{ for some }Q\in\operatorname{Ass}(R/I)=\operatorname{Min}(R/I).$$

The conjecture proposes that adjoining separated stars preserves a precise correspondence between the associated primes of the edge ideal and those of the resulting quotient. The preceding results establish analogous statements for minimal primes and provide computational motivation, while examples show that the corresponding minimal-prime converse can fail even when the extended prime remains associated.