line graphs associated to annihilating-ideal graph attached to lattices of genus one

Let $(l,wedge,vee)$ be a lattice with a least element $0$. the annihilating-ideal graph of $l$, denoted by $mathbb{ag}(l)$, is a graph whose vertex-set is the set of all non-trivial ideals of $l$ and, for every two distinct vertices $i$ and $j$, the vertex $i$ is adjacent to $j$ if and only if $iwedge j={0}$. in this paper, we characterize all lattices $l$ whose the graph $mathfrak{l}(mathbb{ag}(l))$ is toroidal.