Garamvölgyi Dániel: Vertex-connectivity, rigidity and sparsity

Rigidity and global rigidity of graphs are central notions in combinatorial rigidity theory. After recalling their definitions, I will describe various connections between d-rigidity, global d-rigidity, and d-connectivity. It is well-known that d-rigidity has an underlying matroid, which we call the d-dimensional generic rigidity matroid. While there is no matroid for global rigidity, I will explain that it has a "polymatroid-like" structure. I will also show how these combinatorial structures can be used to prove sparsity results for minimally d-connected and minimally globally d-rigid graphs.