Star-critical Ramsey numbers

The graph Ramsey numberR(G,H) is the smallest integer r such that every 2-coloring of the edges of K_r contains either a red copy of G or a blue copy of H. We find the largest star that can be removed from K_r such that the underlying graph is still forced to have a red G or a blue H. Thus, we introduce the star-critical Ramsey numberr_∗(G,H) as the smallest integer k such that every 2-coloring of the edges of K_r−K_1,r−1−k contains either a red copy of G or a blue copy of H. We find the star-critical Ramsey number for trees versus complete graphs, multiple copies of K₂ and K₃, and paths versus a 4-cycle. In addition to finding the star-critical Ramsey numbers, the critical graphs are classified for R(T_n,K_m), R(nK₂,mK₂) and R(P_n,C₄).