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?