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A graph is connected-homogeneous if any isomorphism between finite connected induced subgraphs extends to an automorphism of the graph. In this paper we classify the countably infinite connected-homogeneous graphs. We prove that if Γ is connected countably infinite and connected-homogeneous then Γ is isomorphic to one of: Lachlan and Woodrow's ultrahomogeneous graphs; the generic bipartite graph; the bipartite 'complement of a complete matching'; the line graph of the complete bipartite graph Kdoi:10.1016/j.jctb.2009.04.002 fatcat:unuaac5ywzeazdmf32cd3yqwjy