Redundant edges in Ramsey graphs

For graphs $G$, $F$ and $H$, let $G\to (F,H)$ signify that any edge coloring of $G$ in red and blue contains a red $F$ or a blue $H$. The Ramsey number $R(F,H)=\min \{r\mathrm{\mid }{K}_r\to (F,H)\}$. In this note, we consider redundant edges in Ramsey graphs, which are associate with critical Ramsey numbers. For an integer $k\ge 1$, let $\mathbb{G}=\{G_k,G_{k+1},\dots \}$ be a class of graphs with $\delta (G_n)\ge 1$. We define the critical Ramsey number $R_{\mathbb{G}}(F,H)$ with respect to $\mathbb{G}$ to be $\max{n|; K_r\setminus G_n \rightarrow(F,H),,G_n\in{\mathbb G}\big}$,where$r=R(F,H)$. We shall determine some $R_{\mathbb{G}}(F,H)$, where $\mathbb{G}$ consists of stars, matchings and complete graphs, respectively.