Recwntly I was playing this game, which a graph is presented and the player asked to solve it by moving vertices until no intersections is possible
From the related wikipedia link and the development blog, I learnt about how the graph with no intersection can always be guareenteed to exist if the given graph is genrated in some way, how the game generate its graphs and a paper that mentioned how an algorithm can be used to solve any graphs generated by this game
However I am curious on what's the underlying (or closest) mathematical term that describe my action that is used to solve the graph, because when playing through the levels shown (white circles), I always found I am pulling vertices in a manner very similar to what is shown below
and this caused me to suspect there is some consistent pattern I have been unconsciously applying in order to solve the graphs
From an algorithmic point of view, it's usual to think about where you place new vertices, not how you move them.
Imagine that the UI allowed you to select a vertex, and click on a new position for it. This would make it clearer that the important thing is which side of a set of lines a vertex lies.
For an even clearer way of looking at it, consider the ordering of the lines around each vertex. For three lines - labelled (1, 2, 3) - you can either have (1, 2, 3) or (1, 3, 2) if you follow them round clockwise ... or anticlockwise, you have to pick one; just be consistent.
Now, the problem becomes choosing for each vertex the cyclic ordering of the lines (edges). This is the problem of finding an embedding for the graph - if one exists.
edit:
For example, consider the following graphs -
The top panel shows a drawing with crossings (A), the same embedding but without crossings (B), and an alternative embedding. Note that both A and B have the same clockwise ordering around the top left vertex (1, 2, 3) while C has the ordering (1, 3, 2). Moving the light grey vertex from A->B involves crossing a line, but does not alter the cyclic edge ordering. Moving from B->C does alter the ordering.
For the bottom panel, the two drawings (X and Y) of the same graph are related by a 'Whitney flip'. The clockwise ordering has changed for the top vertex (along others) from (1, 2, 3, 4) to (1, 3, 2, 4) but we haven't moved any vertices across lines.
So this is probably more evidence that the important thing is the ordering of the edges, and not the movement of the vertices. That doesn't really answer the question for the game Planarity, of course!