  A 2-join is an edge cutset that naturally appears in decomposition of several
classes of graphs closed under taking induced subgraphs, such as perfect graphs
and claw-free graphs. In this paper we construct combinatorial polynomial time
algorithms for finding a maximum weighted clique, a maximum weighted stable set
and an optimal coloring for a class of perfect graphs decomposable by 2-joins:
the class of perfect graphs that do not have a balanced skew partition, a
2-join in the complement, nor a homogeneous pair. The techniques we develop are
general enough to be easily applied to finding a maximum weighted stable set
for another class of graphs known to be decomposable by 2-joins, namely the
class of even-hole-free graphs that do not have a star cutset.
  We also give a simple class of graphs decomposable by 2-joins into bipartite
graphs and line graphs, and for which finding a maximum stable set is NP-hard.
This shows that having holes all of the same parity gives essential properties
for the use of 2-joins in computing stable sets.
