Characterization of the stability number of connected graphs

Let $G$ be a connected graph. Write $\gamma_{st}(G)$ for its strong domination number, $\operatorname{st}_{\gamma_{st}}(G)$ for the stability value of $\gamma_{st}$, and $\kappa(G)$ for its vertex connectivity. A vertex set is called critical for maintaining $\gamma_{st}$ when its removal changes the strong domination number as specified by the stability value.

Stability characterization conjecture. For every integer $p\ge 1$,

$$\operatorname{st}_{\gamma_{st}}(G)=p+1$$

if and only if all of the following hold:

1. $\kappa(G)\ge p$;
2. $G$ has at least $p+1$ vertices whose removal does not change $\gamma_{st}$;
3. every set of $p$ vertices is contained in some minimum strong dominating set of $G-S$ for some set $S$ with $|S|\le p$; and
4. there exists a set of $p+1$ vertices that is critical for maintaining $\gamma_{st}$.

The conjecture proposes a structural characterization of graphs with prescribed stability value. The source states that the problem of characterizing graphs with a given stability value remains open.