The Complexity of Drawing a Graph in a Polygonal Region - Lubiw, Anna; Miltzow, Tillmann; Mondal, Debajyoti (Journal of Graph Algorithms and Applications, 2022)
We prove that the following problem is complete for the existential theory of the reals:Given a planar graph and a polygonal region, with some vertices of the graph assigned to points on the boundary of the region, place the remaining vertices to create a planar straight-line drawing of the graph inside the region.This establishes a wider context for the NP-hardness result by Patrignani on extending partial planar graph drawings.Our result is one of the first showing that a problem of drawing planar graphs with straight-line edges is hard for the existential theory of the reals.The complexity of the problem is open in the case of a simply connected region.We also show that, even for integer input coordinates, it is possible that drawing a graph in a polygonal region requires some vertices to be placed at irrational coordinates. By contrast, the coordinates are known to have bounded bit complexity for the special case of a convex region, or for drawing a path in any polygonal region.In addition, we prove a Mnëv-type universality result - loosely speaking, that the solution spaces of instances of our graph drawing problem are equivalent, in a topological and algebraic sense, to bounded algebraic varieties.