## On Testing Asymmetry in the Bounded Degree Graph Model

#### Webpage for a paper by Oded Goldreich

#### Abstract

We consider the problem of testing asymmetry
in the bounded-degree graph model, where a graph is called
asymmetric if the identity permutation is its only automorphism.
Seeking to determine the query complexity of this testing problem,
we provide partial results.
Considering the special case of $n$-vertex graphs
with connected components of size at most $s(n)=\Omega(\log n)$,
we show that the query complexity of $\epsilon$-testing asymmetry
(in this case) is at most $O({\sqrt n}\cdot s(n)/\epsilon)$,
whereas the query complexity of $o(1/s(n))$-testing asymmetry
(in this case) is at least $\Omega({\sqrt{n/s(n)}})$.
In addition, we show that testing asymmetry
in the dense graph model is almost trivial.

#### Material available on-line

- First version posted:
July 2020.
- Revisions: none yet.

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