Lattice representation of undirected graph

Under what circumstances can an undirected graph be represented by integer lattice points in Cartesian coordinates? Specifics:

% Each point on the graph is mapped to (x,y) on the Cartesian grid where 开发者_C百科both x and y are integers.

% Two points (x1,y1) and (x2,y2) on the Cartesian grid are "connected" if abs(x1-x2)<=1 and abs(y1-y2)<=1. In other words, each point has 8 neighbors.

% Two points on the Cartesian graph representation should be connected iff there is an edge between those two points on the graph.

Examples:

% K4: All points are connected to each other.

 
12 
34 

% K2,2: 1 and 2 both connect to both 3 and 4 but there are no other connections.

 
 3 
1 2 
 4 

Since I can't find a lattice representation for K3,2 I'm guessing lattice-able graphs are a proper subset of planar graphs.

Same question for 3D lattice points.

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This lattice type graph is planar since K5 and K3,3 can't be represented. K5 since in K1,5 there is at least one pair of nodes (from 5) that are not connected. K3,3 since it is not possible to place 3 unconnected points that there 'circles' intersect on more that one point.

It is proper subset of planar graphs since K1,9 can't be represented.

For 3D case it is not planar since K5 can be represented, K4 from 2D and one point on top of them. But some planar graphs can't be represented, e.g. K1,28.