The graph Ramsey number R(G, H) is the smallest integer n such that every 2-coloring of the edges of $K_n$ contains either a red copy of G or a blue copy of H and there exists a critical 2-coloring of $K_{n-1}$ that does not contain a red copy of G or a blue copy of H. What is the largest star $K_{1,k}$ that can be removed from $K_n$ so that the underlying graph is still forced to have either a red copy of G or a blue copy of H? That is, determine the largest integer k such that every 2-coloring of $K_n - K_{1,k}$ has either a red G or a blue H and there exists a 2-coloring of $K_n - K_{1,k+1}$ without a red G or a blue H. We have determined this integer for various classes of graphs G and H where R(G, H) is known. For these Ramsey numbers, we have also classified the critical 2-colorings.