On the reduction of Yutsis graphs

General angular momentum recoupling coefficients can be expressed as a summation formula over products of 6-$j$ coefficients. Yutsis, Levinson and Vanagas developed graphical techniques for representing the general recoupling coefficient as a cubic graph and they describe a set of reduction rules allowing a stepwise generation of the corresponding summation formula. This paper gives an overview of the state-of-the-art heuristic algorithms, used in the latest version of our GYutsis program, for calculating general recoupling coefficients. By means of an experimental setup we show that, in particular for problems of higher order, this approach yields summation formulae which are at least as good, but are often more concise than those obtained by previous algorithms. We also give a counter-example showing that the widespread convention of reducing girth cycles first does not always lead to a shortest reduction.