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# Conclusion

**Possible directions:**

> {pause carousel #beyond-expansion}
> ----
> > - Beyond bounded expansion
> ---
> > - Beyond bounded expansion: bounded degeneracy, nowhere dense graphs.
> ---
> > - Beyond bounded expansion: [~~bounded degeneracy~~]{style="color:red"}, nowhere dense graphs.
>
> {focus="sparse-hierarchy"}
>
> {change-page="sparse-hierarchy"}
>
> {unfocus change-page="beyond-expansion" up="conclusion"}
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> {change-page="beyond-expansion"}
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> {pause carousel #beyond-freeness}
> ----
> > - Beyond $H$-freeness
> ---
> > - Beyond $H$-freeness: monotone graph properties.
> ---
> > - Beyond $H$-freeness: monotone graph properties.
> >
> > Esperet and Norin (2022): monotone graph properties are testable on graph classes excluding a minor.
> ---
> > - Beyond $H$-freeness: monotone graph properties.
> >
> > Esperet and Norin (2022): monotone graph properties are testable on graph classes excluding a minor.
> >
> > Goldreich and Ron (2002): bipartiteness is not testable on graphs of bounded expansion.
> ---
> > - Beyond $H$-freeness: [~~monotone graph properties~~]{style="color:red"}.
> >
> > Esperet and Norin (2022): monotone graph properties are testable on graph classes excluding a minor.
> >
> > Goldreich and Ron (2002): bipartiteness is not testable on graphs of bounded expansion.
>
> {change-page="beyond-freeness"}
>
> {change-page="beyond-freeness"}
>
> {change-page="beyond-freeness"}
>
> {change-page="beyond-freeness"}

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**Alternative proof:**

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- Awofeso, Greaves, Lachish, Levi, and Reidl (2025): more **involved** but **finer** analysis.