Rigid Components in Molecular Graphs

Abstract

We consider three-dimensional generic bar-and-joint realizations of squares of graphs. These graphs are also called molecular graphs due to their importance in the study of flexibility in molecules. The Molecular Conjecture, posed in 1984 by T.-S. Tay and W. Whiteley, indicates that determining rigidity (or, more generally, computing the number of degrees of freedom) of molecular graphs may be tractable by combinatorial methods. We show that the truth of the Molecular Conjecture would imply an efficient algorithm to identify the maximal rigid subgraphs of a molecular graph. In addition, we prove that the truth of two other conjectures in combinatorial rigidity (due to A. Dress and D. Jacobs, respectively) would imply the truth of the Molecular Conjecture.