Countable connected-homogeneous graphs

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 Woodrowʹs ultrahomogeneous graphs; the generic bipartite graph; the bipartite ‘complement of a complete matching’; the line graph of the complete bipartite graph K ℵ 0 , ℵ 0 ; or one of the ‘treelike’ distance-transitive graphs X κ 1 , κ 2 where κ 1 , κ 2 ∈ N ∪ { ℵ 0 } . It then follows that an arbitrary countably infinite connected-homogeneous graph is a disjoint union of a finite or countable number of disjoint copies of one of these graphs, or to the disjoint union of countably many copies of a finite connected-homogeneous graph. The latter were classified by Gardiner (1976). We also classify the countably infinite connected-homogeneous posets.