Computational Complexity
Constructing the automorphism group is at least as difficult (in terms of its computational complexity) as solving the graph isomorphism problem, determining whether two given graphs correspond vertex-for-vertex and edge-for-edge. For, G and H are isomorphic if and only if the disconnected graph formed by the disjoint union of graphs G and H has an automorphism that swaps the two components.
The graph automorphism problem is the problem of testing whether a graph has a nontrivial automorphism. It belongs to the class NP of computational complexity. Similar to the graph isomorphism problem, it is unknown whether it has a polynomial time algorithm or it is NP-complete. There is a polynomial time algorithm for solving the graph automorphism problem for graphs where vertex degrees are bounded by a constant (Luks 1982). It is known that the graph automorphism problem is polynomial-time many-one reducible to the graph isomorphism problem, but the converse reduction is unknown.
Read more about this topic: Graph Automorphism
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