I have a graph where each element belongs to one of the two groups $A$ and $B$. An element of group $A$ can have link only with elements from $B$ and vice-versa. I want to ensure the full connectivity of the graph. Now, let $A_i$ denote the set of neighbors of the $i^{th}$ element of A and $B_i$ denote the set of neighbors of the $i^{th}$ element of B. Then, to ensure the full connectivity of the graph, I came up with the following two conditions:
- $\bigcup_i A_i = B$
- $|A_i \cap A_k| > 1$, for all $k \neq i$ where $|\cdot|$ denotes the cardinality of the set.
I wanted to know if these conditions are sufficient to ensure the full connectivity of the graph (or if you have suggestions of other conditions).