An algebraic geometry approach to three dimensional graph rigidity

This report analyses a special topic in the field of graph theory: graph rigidity. Specifically, three dimensional graph rigidity, for which necessary and sufficient conditions, which at the same time are practically workable, still are to be found. This issue corresponds to the graph rigidity problem being conjectured as an NP-hard problem. We will define graph rigidity and flexibility, both locally and globally. We will then create two specific maps acting on the nodes of the graph. This map will yield a system of an equation and an inequality which will describe the rigidity conditions of a given graph. Last, using tools available in Wolfram Mathematica, conclusions on the local/global rigidity of a graph will be drawn based on the output of the algorithm.