  In a series of four papers we determine structures whose existence is dual,
in the sense of complementary, to the existence of stars or combs. Here, in the
third paper of the series, we present duality theorems for a combination of
stars and combs: undominated combs. We describe their complementary structures
in terms of rayless trees and of tree-decompositions.
  Applications include a complete characterisation, in terms of normal spanning
trees, of the graphs whose rays are dominated but which have no rayless
spanning tree. Only two such graphs had so far been constructed, by Seymour and
Thomas and by Thomassen. As a corollary, we show that graphs with a normal
spanning tree have a rayless spanning tree if and only if all their rays are
dominated.
