Partition mesh into predetermined submeshes

For your first question, constructing the adjacency graph of the "partitions" (what you call "cell groups"): Let's say you have an array $p_K$ in which you store for each cell $K$ which partition $p$ it belongs to. Also assume that you have a (sparse) array $a_{KL}$ whose entries are true if cells $K$ and $L$ are neighbors ("adjacent"). Then partition $s$ is "adjacent" to partition $t$ if $t$ is in the set $$ left( bigcup_{K, p_K=s} ; bigcup_{L, a_{KL}=text{true}} left{p_Lright} right)setminus {s}. $$ (In other words, we loop over all cells $K$ in partition $s$, then over all neighbors $L$ of these $K$, and collect their partition indices. This union contains all partition indices that partition $s$ is adjacent to, but will likely include $s$ itself -- so we throw away $s$ at the end.

For your second question, visualization -- this is typically done in the following way:

• For each partition $s$, compute its center of mass $mathbf x_s$, for example by taking the average of the centers of the cells in this partition.
• Compute the center of mass of the entire domain $mathbf {hat x}$, for example by taking the average of the centers of all cells.
• When visualizing the mesh, you draw each triangle with an offset $alpha (mathbf x_s - mathbf {hat x})$ with a small $alpha$ -- say, $alpha=0.05$. In essence, what this does is move each partition outward from the global center of mass by a distance that is proportional to how far that partition already is from the center of mass. Some visualization programs can already do that for you -- the option is typically called "explode".