Countable connected-homogeneous graphs

R. Gray, D. Macpherson
2010 Journal of combinatorial theory. Series B (Print)  
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 K
more » ... 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 . We also classify the countably infinite connectedhomogeneous posets.
doi:10.1016/j.jctb.2009.04.002 fatcat:unuaac5ywzeazdmf32cd3yqwjy