How to create a special graph?

Question

I wish to generate an undirected lattice graph G with following features:

1. It has mxn vertexes;
2. Each vertex has only two kinds of labels (0 or 1);
3. Every txt subgraph is unique.

For example, if the array of the vertex label of lattice graph G is

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

then any 3x3 subgraph of G is unique (only one isomorphism can be found). For example, subgraph

1 0 1
0 1 1
0 0 1

can only be the top-left block of G, 90 degree rotated.

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Actually, I can generate such graphs by wave function collapsing. The example above is generated in this way.

However, I want to go further. Here are the true questions:

1. If the subgraph can be any shape (with at least t vertex), how to generate the graph G?
2. If graph G is not limited in lattice shape, what will happen?
3. What are the relationships between the shapes of subgraph, and the minimum vertex number t to make them unique?

Any idea or keyword?