# Types of symmetry for combinatorial graphs

Let $G$ be an undirected, connected graph without loops. Let's call $G$ symmetric iff it has a non-trivial automorphism (that is a permutation $\pi : V(G) \rightarrow V(G)$ – which is not the identity permutation – that preserves the edge relation). Further, let us denote the set of fix points of a permutation $\pi$ with $FP(\pi)$

Let's say $G$ is point-symmetric if the following applies:

• $G$ is symmetric
• Let $\Pi$ be the set of all non-trivial permutations which constitute an automorphism for $G$
• Let $V_p = \bigcap_{\pi \in \Pi} FP(\pi)$ , so with other words all the vertices which can't be swapped with some other vertex for an automorphism
• $V_p$ is non-empty

And let's say $G$ is bi-symmetric if the following holds:

• $G$ is symmetric and $|V(G)|$ is even
• There exists a subset $V_b \subset V(G)$ with $2*|V_b| = |V(G)|$
• The vertex-induced subgraph $G[V_b]$ is connected
• $G[V_b]$ is isomorph to $G[V(G) \setminus V_b]$

Or maybe put more simply, $G$ contains two parts which can be swapped to obtain an automorphism.

I just thought of these two kinds of symmetry for which I think they are described in group theory(maybe as special cases of a more generalized concept) but due to my limited knowledge in this field I wouldn't know where to look. I'm interested in properties of these both types of symmetries.

So my question is: under what kind of name are these concepts of symmetry known or what would be the name of a broader concept containing them?