Let $G$ be a connected, undirected graph. We define a block $B$ to be a maximal $2$-connected induced subgraph in $G$. It is easy to see that any two distinct blocks are either disjoint or overlap at a single vertex (called a "cut-vertex"). Moreover, it is well known that the blocks of $G$ and their cut vertices form a tree structure.
Now, switch gears and let $G$ be a strongly connected, directed graph. Suppose we define a "block" $B$ in this setting to be a maximal $2$-strongly connected induced subgraph in $G$. That is, $B$ is a block in $G$ if it is a maximal subset of vertices in $G$, with the property that for any distinct $s, t\in B$, there are two disjoint paths from $s$ to $t$ in the induced subgraph on $B$ (note that we can swap the roles of $s$ and $t$, so that two disjoint paths from $t$ to $s$ must also exist in $B$).
My Question: In this directed setting, do the blocks (and perhaps some equivalent notion of cut-vertices) also admit a tree structure (analogous to the undirected case)? If not, do they at least admit some interesting structure?
If found, I'd also appreciate any references to the literature on obtaining some sort of block-cut decomposition for directed graphs.
