An analogue of Reidemeister moves for plane graphs It's well-known that if two link diagrams represent the same link, then they can be transformed into each other by a sequence of Reidemeister moves. Is there a similar set of "moves" on plane graphs that transform two embeddings of a planar graph into each other?
I am looking at correspondences between link diagrams and their medial graphs and I was hoping to get a correspondence between links and planar graphs.
 A: A theorem of Whitney says that a $3$-connected planar graph has a unique embedding into the $2$-sphere $S^2$, up to composition with a homeomorphism of $S^2$. 
Distinct embeddings of a planar graph $G$ into $S^2$ are related by a sequence of $1$-flips and $2$-flips, or Whitney flips. I will describe both of these operations below. My information comes from this arXiv paper which references this result without a proof. They refer to the book "Graphs on Surfaces" by Mohar and Thomassen for the complete proof.
1-flip: Let $G$ be a graph with cut vertex $v$ with given planar embedding. A $1$-flip of $G$ is obtained by splitting the cut vertex into two vertices $v_1$ and $v_2$ resulting in two components in the graph $G_1$ and $G_2$, and then gluing $v_1$ and $v_2$ back together to yield a new embedding.
2-flip: Let $\{u,v\}$ be a set of vertices of $G$ whose deletion disconnects the graph. A 2-flip is obtained by splitting $u$ into two vertices, splitting $v$ into two vertices, rotating the one of the two resulting components, then gluing $u$ and $v$ back together. 
Examples of $1$-flips and $2$-flips are below.

On knots and links. Each planar graph gives rise to a medial graph that can be interpreted as a link projection. A common convention is to assign crossings to the link diagram so that it is alternating. This leaves two choices for the link diagram assigned to graph (a link diagram and its mirror image). It is not hard to devise a convention to pick one of these two diagrams. 
Unfortunately, $2$-flips that preserve the isomorphism class of $G$ but change its planar embedding do not always preserve the isotopy class of the link. In particular, these $2$-flips can lead to non-equivalent mutant links. 
If we desire a simple diagrammatic move that preserves the isotopy class of a link, then we need to consider flypes. The Tait flyping conjecture (proven by Menasco and Thistlethwaite) states that any two alternating diagrams of the same link are related by a sequence of flypes, the operation pictured below.

A flype on a link diagram induces a special type of $2$-flip on its checkerboard graph. So equivalence up to these types of $2$-flips will give the same medial links. 
If we'd like to expand to consider nonalternating link diagrams, then the graphs we consider must have signed edges. The signed edges tell us about the crossing information on the medial graph/link. No analogue of the Tait flyping conjecture exists for nonalternating links, and so the story here is much more complicated.
