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This article assumes that all graphs are finite. For \$s\geq 1\$ and a graph \$\G\$, for every pair of isomorphic connecting induced subgraphs on at most \$s\$ vertices, the self of \$\G\$ that maps the first vertex to the second If an isomorphism exists, say: If \$\G\$ is \$s\$-connected-set-homogeneous and all isomorphisms between two isomorphic-connected induced subgraphs over at most \$s\$ vertices can be extended to automorphisms of \$\G\$, \$\ G\$ is \$s\$-connected-homogeneous. For \$n\geq 1\$ the graph \$\G\$ is locally said to be \$2\K_n\$[\G(u)]\$ is induced to the set of vertices in \$\G\$ adjacent to a given vertex \$u\$ and is isomorphic to \$2\K_n\$.

Note that \$2\$-connected-set-homogeneous is not a \$2\$-connected-homogeneous graph. I posed the problem of characterizing or classifying \$-connected-set-homogeneous graphs (Eur. J. Combin. 93 (2021) 103275). Until now, there are only two known families of \$3\$-connected-set-homogeneous graphs of surrounding \$3\$ that are not \$3\$-connected-homogeneous, and these graphs are locally \$2\K_n\$ and \$n =2\$ or \$4. In this paper, we complete a classification of finite \$3\$-connected-set-homogeneous graphs that are locally \$2\K_n\$ and \$n\geq 2\$ , and that every such graph has some specific \$2\$ Line graph for -arc-transitive. graph. Furthermore, it provides a good description of the finite \$3\$-connected-set-homogeneous graph, which is locally \$2\K_n\$ and not the \$3\$-connected-set-homogeneous graph with a solvable automorphism group. increase. This is used to build some new \$3\$-connected-set-homogeneous but \$3\$-connected-homogeneous graphs and some new \$2\$-arc-transitive graphs.

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