  This paper is the second in a series in which the authors study the conjugacy
decision problem (CDP) and the conjugacy search problem (CSP) in Garside
groups. The ultra summit set USS(X) of an element X in a Garside group G is a
finite set of elements in G, introduced by the second author, which is a
complete invariant of the conjugacy class of X in G. A fundamental question, if
one wishes to find bounds on the size of USS(X), is to understand its
structure. In this paper we introduce two new operations on elements of USS(X),
called `partial cycling' and `partial twisted decycling', and prove that if Y
and Z belong to USS(X), then Y and Z are related by sequences of partial
cyclings and partial twisted decyclings. These operations are a concrete way to
understand the minimal simple elements which result from the convexity theorem
in the mentioned paper by the second author. Using partial cycling and partial
twisted decycling, we investigate the structure of a directed graph \Gamma_X
which is determined by USS(X), and show that \Gamma_X can be decomposed into
`black' and `grey' subgraphs. There are applications which relate to the
program, outlined in the first paper in this series, for finding a polynomial
solution to the CDP/CSP in the case of braids. A different application is to
give a new algorithm for solving the CDP/CSP in Garside groups which is faster
than all other known algorithms, even though its theoretical complexity is the
same as that given by the second author. There are also applications to the
theory of reductive groups.
