A graph Γ is k-connected-homogeneous (k-CH) if k is a positive integer and any isomorphism between connected induced subgraphs of order at most k extends to an automorphism of Γ, and connected-homogeneous (CH) if this property holds for all k. Locally finite, locally connected graphs often fail to be 4-CH because of a combinatorial obstruction called the unique x property; we prove that this property holds for locally strongly regular graphs under various purely combinatorial assumptions. We then classify the locally finite, locally connected 4-CH graphs. We also classify the locally finite, locally disconnected 4-CH graphs containing 3-cycles and induced 4-cycles, and prove that, with the possible exception of locally disconnected graphs containing 3-cycles but no induced 4-cycles, every finite 7-CH graph is CH.

如果k是一个正整数，并且图的Γ是k个同构的（k -CH），并且连通诱导子图之间最多k个同构子图的任何同构都可以扩展为Γ的自同构，如果该性质成立，则它是同构的（CH）对于所有k。由于称为唯一x的组合障碍，局部有限，局部连接的图通常无法成为4-CH属性; 我们证明了该性质适用于各种纯粹组合假设下的局部强正则图。然后，我们对局部有限，局部连接的4-CH图进行分类。我们还对包含3个循环和诱导4个循环的局部有限，局部断开的4-CH图进行分类，并证明，除了包含3个循环但不包含诱导4个循环的局部断开的图之外，每个有限7- CH图是CH。