## The complexity of testing all properties of planar graphs, and the role of isomorphism

by Sabyasachi Basu, Akash Kumar, and C. Seshadhri

In the past, some people collected mail stamps, others collected other items. Nowadays, it seems natural to collect ideas (or notions). Anyhow, I collect complexities that characterize the requirements for natural computational problems. So here is one to my collection. [Myself, in post Nr 277]
Needless to say, I am even happier to add to my collection an item that represents a natural property testing problem. Specifically, the new item refers to the complexity of testing, in the bounded-degree graph model, whether a given graph is isomorphic to a fixed graph. The current paper shows that this complexity is exponential in the square of the reciprocal of the proximity parameter. Furthermore, it is shown that this complexity is maximal among all properties of planar graphs.

#### The original abstract

Consider property testing on bounded degree graphs and let $\varepsilon > 0$ denote the proximity parameter. A remarkable theorem of Newman-Sohler (SICOMP 2013) asserts that all properties of planar graphs (more generally hyperfinite) are testable with query complexity only depending on $\varepsilon$. Recent advances in testing minor-freeness have proven that all additive and monotone properties of planar graphs can be tested in $\poly(\varepsilon^{-1})$ queries. Some properties falling outside this class, such as Hamiltonicity, also have a similar complexity for planar graphs. Motivated by these results, we ask: can all properties of planar graphs can be tested in $\poly(\varepsilon^{-1})$ queries? Is there a uniform query complexity upper bound for all planar properties, and what is the hardest" such property to test?

We discover a surprisingly clean and optimal answer. Any property of bounded degree planar graphs can be tested in $\exp(O(\varepsilon^{-2}))$ queries. Moreover, there is a matching lower bound, up to constant factors in the exponent. The natural property of testing isomorphism to a fixed graph requires $\exp(\Omega(\varepsilon^{-2}))$ queries, thereby showing that (up to polynomial dependencies) isomorphism to an explicit fixed graph is the hardest property of planar graphs. The upper bound is a straightforward adapation of the Newman-Sohler analysis that tracks dependencies on $\varepsilon$ more carefully. The main technical contribution is the lower bound construction, which is achieved by a special family of planar graphs that are all mutually far from each other.

We can also apply our techniques to get analogous results for bounded treewidth graphs. We prove that all properties of bounded treewidth graphs can be tested in $\exp(O(\varepsilon^{-1}\log \varepsilon^{-1}))$ queries. Moreover, testing isomorphism to a fixed forest requires $\exp(\Omega(\varepsilon^{-1}))$ queries.

Available from ECCC TR21-122

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