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.
