TITLE: is there a ‘nice’ lattice on the set of unlabelled graphs with $n$ vertices?
QUESTION [10 upvotes]: It is easy to endow the set of vertex-labelled graphs with $n$ vertices with a lattice structure: take the union and the intersection of the edge set as meet and join respectively.
However, I wonder whether there is a ‘nice’ lattice structure on unlabelled graphs, i.e., graphs up to isomorphism.
I'd also be happy with a lattice structure on a (reasonably) large subset, if this makes things any easier, e.g., connected or planar graphs on $n$ vertices.
To clarify: a very non-‘nice’ way to make the set of unlabelled graphs on $n$ vertices into a lattice is to pick an arbitrary total order.
An idea for a slightly nicer way might be to set $G < H$ if and only if $H$ has more edges than $G$, however this does not produce a lattice.

REPLY [2 votes]: I guess this class is way too small, but since the questions is still unanswered, I will give you an example I like.
An EFG (Edge Firing Game) is defined from an undirected graph $G$ with a distinguished vertex called the sink. It starts from an orientation of $G$ and generates a set $L$ of orientations as follows. The initial orientation is in $L$, and if an orientation is in $L$ and has a vertex (other than the sink) with only incoming edges, then the orientation obtained by reversing all these edges also is in $L$.
                      
This operation defines a relation between the orientations in $L$, see above example (the sink is the black square); it is the covering relation of a distributive lattice. This was shown by James Propp in 1993 together with many other interesting results on lattices and orientations.
Interestingly, this game actually encodes all distributive lattices: each of them may be obtained from an EFG, which we have shown in 2001.