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, eg. 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.

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.

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, eg. 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.

A slightly nicer way might be to set $G < H$ if and only if $H$ has more edges than $G$. Note that the second construction will not work for connected graphs, because there would not be a bottom element.

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, eg. 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.

It is easy to endow the set of vertex-labelled graphs with $n$ vertices with a lattice structure. However, I wonder whether there is a 'nice' lattice structure on unlabelled graphs.

I'd also be happy with a lattice structure on a (reasonably) large subset, if this makes things any easier, eg. connected or planar graphs on $n$ vertices.

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, eg. 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.

A slightly nicer way might be to set $G < H$ if and only if $H$ has more edges than $G$. Note that the second construction will not work for connected graphs, because there would not be a bottom element.