Combinatorial aspects of spatial frameworks

The rigidity or flexibilty of a skeletal structure might be investigated by asking questions about its underlying graph.
While combinatorial criteria are known to determine whether a framework is rigid by knowledge about its underlying graph,
provided the graph is embedded on a line or in the plane, a combinatorial criteria to determine the rigidity of spatial frameworks is not at hand.
Isostatic graphs (the underlying graphs of rigid frameworks which become flexible if an arbitrary bar is removed from the framework)
have several interesting properties and the problem of finding combinatorial criteria for a framework to be rigid reduces to the problem of finding combinatorial criteria for graphs to be isostatic.

So called abstract rigidity matroids present a matroid approach to rigidity theory.

Alternative characterizations of this family of matroids in terms of bases, circuits and hyperplanes of these matroids are presented.

Afterwards, framework decompositions are introduced, which generalize 3T2-decompositions of graphs of 2-isostatic frameworks to higher dimensions.
We show that it is sufficient for a graph to admit a proper spatial framework decomposition to be 3-isostatic.
After discussing some properties of graphs with such decompositions, we present an algorithm that finds such proper decompositions for planar 3-isostatic graphs.