Formation control deals with the design of decentralized control laws that stabilize mobile, autonomous agents at prescribed distances from each other. We call any configuration of the agents a target configuration if it satisfies the inter-agent distance conditions. It is well known that when the distance conditions are defined by a rigid graph, there is a finite number of target configurations modulo rotations and translations of the entire formation. We can thus recast the objective of formation control as stabilizing one or many of the target configurations. A major issue is that such control laws will also have equilibria corresponding to configurations which do not meet the desired inter-agent distance conditions; we refer to these as undesirable configurations. The undesirable configurations become problematic if they are also stable. Designing decentralized control laws whose stable equilibria are all target configurations in the case of a general rigid graph is still an open problem. We provide here a new point of view on this problem, and propose a partial solution by exhibiting a class of rigid graphs and control laws for which all stable equilibria are target configurations.