On the inclusion ideal graph of a poset

Let $(P, \leq)$ be an atomic partially ordered set (poset, briefly) with a minimum element $0$ and $\mathcal{I}(P)$ the set of nontrivial ideals of $ P $. The inclusion ideal graph of $P$, denoted by $\Omega(P)$, is an undirected and simple graph with the vertex set $\mathcal{I}(P)$ and two distinct vertices $I, J \in \mathcal{I}(P) $ are adjacent in $\Omega(P)$ if and only if $ I \subset J $ or $ J \subset I $. We study some connections between the graph theoretic properties of this graph and some algebraic properties of a poset. We prove that $\Omega(P)$ is not connected if and only if $ P = \{0, a_1, a_2 \}$, where $a_1, a_2$ are two atoms. Moreover, it is shown that if $ \Omega(P) $ is connected, then $\operatorname{diam}(\Omega(P))\leq 3$. Also, we show that if $ \Omega(P) $ contains a cycle, then $\operatorname{girth}(\Omega(P)) \in \{3,6\}$. Furthermore, all posets based on their diameters and girths of inclusion ideal graphs are characterized. Among other results, all posets whose inclusion ideal graphs are path, cycle and star are characterized.