when does the complement of the annihilating-ideal graph of a commutative ring admit a cut vertex?

the rings considered in this article are  commutative  with identity which admit at least two  nonzero annihilating ideals. let $r$ be a ring. let $mathbb{a}(r)$ denote the set of all annihilating ideals of $r$ and let $mathbb{a}(r)^{*} = mathbb{a}(r)backslash {(0)}$. the annihilating-ideal graph of $r$, denoted by $mathbb{ag}(r)$  is an undirected simple graph whose vertex set is $mathbb{a}(r)^{*}$ and distinct vertices $i, j$ are joined by an edge in this graph if and only if $ij = (0)$. the aim of this article is to classify rings  $r$ such that $(mathbb{ag}(r))^{c}$ ( that is,  the complement of $mathbb{ag}(r)$)   is connected and admits a cut vertex.