On some path-critical Ramsey numbers

For graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the smallest $r$ such that any red-blue edge coloring of $K_r$ contains a red $G$ or a blue $H$. The path-critical Ramsey number $R_{\pi }(G,H)$ is the largest $n$ such that any red-blue edge coloring of $K_r\setminus P_n$ contains a red $G$ or a blue $H$, where $r=R(G,H)$ and $P_n$ is a path of order $n$. In this note, we show a general upper bound for $R_{\pi }(G,H)$, and determine the exact values for some cases of $R_{\pi }(G,H)$.