Ordinary isogeny graphs over ${\mathbb{F}}_p$: the inverse volcano problem

We give a detailed presentation of $\mathrm{\ell }$-isogeny graphs associated with ordinary elliptic curves defined over ${\mathbb{F}}_p$. We then focus on the following inverse problem: given an abstract volcano $V$, do there always exist primes $\mathrm{\ell },p\in \mathbb{N}$ such that the ordinary $\mathrm{\ell }$-isogeny graph over ${\mathbb{F}}_p$ contains $V$ as a connected component? We provide an affirmative answer to this question.