A comparison of vertex-splitting and spider-splitting for the study of three-dimensional rigidity

Abstract

In two dimensions, generic rigidity is a combinatorial property of a framework, but extensions into three dimensions fail to completely characterize generic rigidity. It is therefore interesting to investigate two graph operations introduced by Walter Whiteley, vertex-splitting and spider-splitting, which are known to take a minimally rigid framework in three dimensions to a new minimally rigid framework with an additional vertex. We present algorithms for generating all possible graphs obtained by vertex-splitting, spider-splitting, and combinations of vertex-splitting and spider-splitting. For graphs with up to and including 8 vertices, the set of graphs obtained by spider-splitting is a subset of the set obtained by vertex-splitting. Additionally, the set produced by combinations of vertex-splitting and spider-splitting is equal to the set obtained by vertex-splitting. This suggests that as a method for generating rigid graphs, spider-splitting is inferior to vertex-splitting at all steps of iteration.