  The fundamental group of a finite graph of groups with trivial edge groups is
a free product. We are interested in those outer automorphisms of such a free
product that permute the conjugacy classes of the vertex groups. We show that
in particular cases of interest, such as where vertex groups are themselves
finite free products of finite and cyclic groups, given such an outer
automorphism, after passing to a positive power, the outer automorphism is
represented by a particularly nice kind of relative train track map called a
CT. CTs were first introduced by Feighn and Handel for outer automorphisms of
free groups. We develop the theory of attracting laminations for and principal
automorphisms of free products. We prove that outer automorphisms of free
products satisfy an index inequality reminiscent of a result of Gaboriau,
Jaeger, Levitt and Lustig and sharpening a result of Martino. Finally, we prove
a result reminiscent of a theorem of Culler on the fixed subgroup of an
automorphism of a free product whose outer class has finite order.
