3x3 Connected Components
J, 54 bytes

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A function taking a list in the order ABCDEFGHI.



Given an adjacency matrix A of a graph of order n, the graph is connected if and only if all the entries of (A + I)n are nonzero, where I is the n×n identity matrix.

We start with the (fixed) adjacency-plus-identity matrix of the 3×3 king grid (in the order ABCDEFGHI,) namely:

1 1 0 1 1 0 0 0 0
1 1 1 1 1 1 0 0 0
0 1 1 0 1 1 0 0 0
1 1 0 1 1 0 1 1 0
1 1 1 1 1 1 1 1 1
0 1 1 0 1 1 0 1 1
0 0 0 1 1 0 1 1 0
0 0 0 1 1 1 1 1 1
0 0 0 0 1 1 0 1 1


.
For each label l, we select the rows and columns corresponding to the nodes of label l.
This gives us the adjacency-plus-identity matrix of the subgraph of l-labeled nodes.
We then use this matrix to test if the subgraph is connected, as described above.

We construct the above matrix by noting that if we let

    0 0 0
Z = 0 0 0
    0 0 0


and

    1 1 0
O = 1 1 1
    0 1 1


, then the matrix can be seen as the 3×3 block matrix

O O Z
O O O
Z O O


, which has the same pattern as O!
O is produced by repeating the pattern 1 1 0 1 in a 3×3 block.