One brick at a time:a survey of inductive constructions in rigidity theory

Nixon, Anthony and Ross, Elissa (2014) One brick at a time:a survey of inductive constructions in rigidity theory. In: Rigidity and symmetry. Fields Institute Communications . Springer, New York, pp. 303-324. ISBN 9781493907809

Abstract

We present a survey of results concerning the use of inductive constructions to study the rigidity of frameworks. By inductive constructions we mean simple graph moves which can be shown to preserve the rigidity of the corresponding framework. We describe a number of cases in which characterisations of rigidity were proved by inductive constructions. That is, by identifying recursive operations that preserved rigidity and proving that these operations were sufficient to generate all such frameworks. We also outline the use of inductive constructions in some recent areas of particularly active interest, namely symmetric and periodic frameworks, frameworks on surfaces, and body-bar frameworks. As the survey progresses we describe the key open problems related to inductions.