Explorations on the number of realizations of minimally rigid graphs

Abstract

Rigid graphs have only finitely many realizations. In the recent years significant progress was made in computing the number of such realizations. With this progress it was also possible for the first time to do computations on large sets of graphs. In this paper we show what we can conclude from the data we got from these computations. This includes new lower bounds on the maximal realization count for a given number of vertices, upper bounds for the minimal realization count in higher dimensions and effects of rigidity preserving construction rules on the realization number. In all cases we give certificate graphs which prove the respective results.