The graph Ramsey number R(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. The star-critical Ramsey number r * (G, H) is 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 will classify the critical graphs, 2-colorings of the complete graph on R(G, H) − 1 vertices with no red G or blue H, for the path-path Ramsey number. This

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... assification will be used in the proof of r * (P n , P m ).