  A graph $\Gamma$ 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 $\Gamma$, 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.
