I'm sorry if the title is a bit convoluted. I'm a bit unsure how to formulate this condition in words, see below instead.
Say we are given a set $Y$. I want to find the following set: $\mathcal{A}$ such that for any $x=\{G_1, ..., G_k\} \in \mathcal{A}, G_i \subset Y$, we have that:
$\forall G_i \in x, \exists G_j \in x, i \neq j , \quad G_i \cap G_j \neq \emptyset $
I'm mostly interesting in finding an algorithm which generates this set. This problem arose when trying to figure out how to generate the set of subgraphs of a connected subgraphs, such that their union is still connected. So the sets above would then be vertex sets and I would form their induced graph.
So, I want to find the subgraphs such that the union of all those subgraphs $\bigcup_{i=1}^{k}G_{i} \text{ is connected}$. Since each of the subgraphs themselves are connected (since the graph is), the above stated condition ought to ensure that their union is connected.
Anyone have an idea how to tackle this? Any help is much appreciated!