When I looked up the definition of convex drawing of planar graph, my confusion mainly focused on the outer face.
The following definition of convex drawing is from Wikipedia.
In graph drawing, a convex drawing of a planar graph is a drawing that represents the vertices of the graph as points in the Euclidean plane and the edges as straight line segments, in such a way that all of the faces of the drawing (including the outer face) have a convex boundary.
It gives two examples
Surprisingly, the outer face is not convex in the second example.
I 'd like to ask what is the convex boundary of outer face here? The outer face doesn't seem to be convex and I feel that the unbounded face is the complement of a convex set.
So I'm confused by the definition of convex drawing where boundary of the outer face is also convex. I don't know where I got it wrong.
I searched the Mathemaics Stack for a similar puzzle below. There seems to be no consensus.
https://math.stackexchange.com/questions/940693/convex-planar-graphs
Best Answer
Wikipedia does not state that the outerface is convex, but rather that it has a convex boundary. I assume this means that the set consisting of all points inside the boundary is convex. So for example, the outerface has a convex boundary in your two examples. Also, with this definition, it is true that every $3$-connected planar graph has a convex drawing (since it is the $1$-skeleton of a $3$-polytope).