  Let $l$ and $p$ be two distinct prime numbers. We study $l$-isogeny graphs of
ordinary elliptic curves defined over a finite field of characteristic $p$,
together with a level structure. Firstly, we show that as the level varies over
all $p$-powers, the graphs form an Iwasawa-theoretic abelian $p$-tower, which
can be regarded as a graph-theoretical analogue of the Igusa tower of modular
curves. Secondly, we study the structure of the crater of these graphs,
generalizing previous results on volcano graphs. Finally, we solve an inverse
problem of graphs arising from the crater of $l$-isogeny graphs with level
structures, partially generalizing a recent result of Bambury, Campagna and
Pazuki.
