I have learnt that every face in a planar graph has sides, and that sides are edges which bound the face clockwise. I am very confused about a few things regarding sides:
- I am not seeing how the single infinite face for a tree graph is bounded by any edges at all- how can an infinite region be bounded, which would imply it is finite?
- I am not seeing why we would count an edge as being in 2 different sides for a undirected tree graph- how does this edge constitute 2 different sides in the boundary? Is it because an undirected graph has 2 directed edges for each edge? So if we replaced every undirected edge in nthe undirected tree graph with a directed edge, we would wouldn't count these edges twice in the degree of the single face?
- What does it mean for the sides to bound the face "clockwise"? What does clockwise mean here? I am not seeing how this notion is relevant as to the definition of a side.
EDIT- it appears the notion of how many sides a face has is related to the notion of the degree of a face, and the latter notion is used instead of the former in certain contexts- so for those who are unfamiliar with the definition I presented, this is how it related to the seemingly more common notion of degree of a face.
It may be easier to understand the terminology, if you think of adding a point at infinity and using the stereographic projection to view the graph as embedded in a sphere rather than the plane. Then the infinite face becomes a finite space that contains the point at infinity. If there are edges which have that face on both sides (as they all will in a tree) then that edge will appear twice when you work around the edges of that face.
As a minimalist example, a graph comprising just a single edge has as its exterior a face that you can think of as a disc with the perimeter divided into two semi-circles corresponding to the two sides of the edge. (Think of cutting a slit in a basketball.)
Note that this only works for connected graphs that have at least one edge.