### Revision(s):

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Revision #2 to TR20-118 | 9th January 2021 11:01
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#### On Testing Asymmetry in the Bounded Degree Graph Model

**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 $O({\sqrt n}\cdot s(n)/\epsilon)$ and at least $\Omega({\sqrt{n^{1-O(\epsilon)}/s(n)}})$. In particular, the query complexity of $o(1/s(n))$-testing asymmetry

is at least $\Omega({\sqrt{n/s(n)}})$.

In addition, we show that testing asymmetry in the dense graph model is almost trivial.

**Changes to previous version:**
Correcting inaccuracies introduced in yesterday's revision.

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Revision #1 to TR20-118 | 8th January 2021 14:53
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#### On Testing Asymmetry in the Bounded Degree Graph Model

**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 $\e$-testing asymmetry (in this case) is at most $O({\sqrt n}\cdot s(n)/\e)$ and at least $\Omega({\sqrt{n^{1-O(\e)}/s(n)}})$. In particular, the query complexity of $o(1/s(n))$-testing asymmetry

is at least $\Omega({\sqrt{n/s(n)}})$.

In addition, we show that testing asymmetry in the dense graph model is almost trivial.

**Changes to previous version:**
The result was generalized to offer lower bounds for any value of the proximity parameter. This is supported by the new Proposition 6.

### Paper:

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TR20-118 | 5th August 2020 00:41
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#### On Testing Asymmetry in the Bounded Degree Graph Model

**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.