In this paper, all graphs are assumed to be finite. Let $s\ge 1$ be an integer. A graph is called $s$-CSH ($s$-connected-set-homogeneous) if for every pair of isomorphic connected induced subgraphs on at most $s$ vertices, there exists an automorphism mapping the first to the second. A graph is called $s$-SH ($s$-set-homogeneous) if for every pair of isomorphic induced subgraphs (not necessarily connected) on at most $s$ vertices, there exists an automorphism mapping the first to the second. A graph is called $s$-homogeneous (respectively $s$-CH, that is, $s$-connected-homogeneous) if every isomorphism between two induced subgraphs (respectively, connected induced subgraphs) on at most $s$ vertices extends to an automorphism of the whole graph.

The first main result, Theorem 1.1, proves that each connected 3-CSH graph is arc-transitive. A consequence of this result is that each 3-CSH graph is 2-CH. Note that 2-CSH but not 2-CH graphs are just half-arc-transitive graphs which have been extensively studied in the literature. Motivated by this, it is natural to consider 3-CSH but not 3-CH graphs. In this paper, we first prove that there exist infinitely many 3-CSH but not 3-CH graphs, and then prove that every prime valent 3-CSH graph is 3-CH. Finally, using these two results, we classify all arc-regular 3-CSH but not 3-CH graphs of girth 3.

在本文中，所有图均假定为有限的。让$s\ge \mathrm{1个}$是一个整数。图称为$s$-CSH（$s$-connected-set-homogeneous），如果最多对每对同构连接的诱导子图$s$顶点，存在自同构映射第一个到第二个。图称为$s$-SH（$s$-set-homogeneous），如果对于每对同构诱导子图（不一定是连接的）最多$s$顶点，存在自同构映射第一个到第二个。图称为$s$-均质的（分别$s$-CH，即$s$-连通同质的），如果最多在两个诱导子图（分别是连通诱导子图）之间的每个同构$s$ 顶点扩展到整个图的自同构。

第一个主要结果（定理1.1）证明了每个相连的3-CSH图都是弧传递的。该结果的结果是每个3-CSH图都是2-CH。请注意，2-CSH图而不是2-CH图只是半弧传递图，已在文献中进行了广泛研究。因此，考虑3-CSH而不考虑3-CH图是很自然的。在本文中，我们首先证明存在无限多的3-CSH图，但没有3-CH图，然后证明每个素价的3-CSH图都是3-CH。最后，使用这两个结果，我们对所有弧形3-CSH进行了分类，但没有对周长3的3-CH图进行分类。