I am given a connected graph $G$ with some embedding. I want to find a non-deterministic algorithm running in $O(n)$ time to decide whether $G$ with that embedding is a plane graph (i.e, can be drawn without intersecting edges).
I know there exist algorithms that decide whether a graph is planar or not in time $O(n)$, but they are not given a fixed embedding, and only check whether there exists an embedding in which the graph is planar or not.
I also know that there are $O(n\log n)$ time algorithms that compute the intersection points of lines, but since my input is a graph with an embedding, it is less "general" than finding the intersection of a set of lines, so I suspect there might be a faster algorithm here.
I would be glad if you could refer me to some sources about this problem, and if a solution exists I would like to know more about it. Thanks in advance!
