Based on comments to my earlier answer:
It seems the graphs are all undirected and planar, i.e. can be embedded in a 2D plane without crossing edges, and one such embedding is given. This embedding will partition the plane. E.g. a figure 8 partitions the plane in three: two "inner" areas and an infinite outer area. An alternative view is that all edges of a node are cyclically ordered. (This is the essential part that allows us to apply graph theory)
A partition is necessarily enclosed by a cycle, but not all cycles may partition a single area. In the trivial case of a figure 8, though, all three areas are directly associated with a distinct cycle.
The input graph can generally be simplified. Some nodes may have only a single edge; they can't contribute to the partitioning and can be removed along with the edge. Other nodes have two edges connecting distinct nodes. Here, the node and the two edges can be replaced by a direct edge connecting the neighbors. I.e. a figure 8 graph can be simplified to two nodes and three edges between them. (This is not a necessary step but helps computation).
Now, each vertex will have two areas to either side (since they're undirected, "left and right" aren't obvious distinctions). So, for |V| vertices, we need to consider 2 * |V| sides. They're in general not distinct. Two adjacent edges (connected to the same node) may border the same area, if they're also adjacent in the cyclic order of edges of that node. Obviously, for nodes with only two edges, the two edges share both areas (which is why we'd eliminated them in the previous step). For nodes with three edges, any two edges share at least one area.
So, here's how to enumerate those areas: Assign a sequential number to all edges and vertices. Assign a direction to each edge so it runs from the lower-numbered edge to the higher. Start with vertex 1, right side, and number this area 1. Trace the boundary edges of this area, assigning the same number 1 to the appropriate sides of its boundary edges. You do this by taking at each node the next adjacent edge in counter-cyclical order. When you get back to your starting point, you know all edges bounding area 1.
You then check the left edge of the first vertex. If it's not part of area 1, then it's area 2, and you apply the same algorithm. Next, check vertex 2, right side and left side, etc. Each time you find an edge and a side that's unnumbered yet, assign the next area number and trace the edges of the newly founded area.
There's a slight problem with determining which area number corresponds to infinity. To see this, take a simple () graph: two edges, two nodes, and two areas (inside and outside). Due to the random numbering of edges and vertices, outside may end up as either 1 or 2. That's unavoidable; in graph theory there's no distinction between inside and outside.
