  This paper introduces the notion of involution module, the first
generalization of the modular decomposition of 2-structure which has a unique
linear-sized decomposition tree. We derive an O(n^2) decomposition algorithm
and we take advantage of the involution modular decomposition tree to state
several algorithmic results. Cographs are the graphs that are totally
decomposable w.r.t modular decomposition. In a similar way, we introduce the
class of switch cographs, the class of graphs that are totally decomposable
w.r.t involution modular decomposition. This class generalizes the class of
cographs and is exactly the class of (Bull, Gem, Co-Gem, C_5)-free graphs. We
use our new decomposition tool to design three practical algorithms for the
maximum cut, vertex cover and vertex separator problems. The complexity of
these problems was still unknown for this class of graphs. This paper also
improves the complexity of the maximum clique, the maximum independant set, the
chromatic number and the maximum clique cover problems by giving efficient
algorithms, thanks to the decomposition tree. Eventually, we show that this
class of graphs has Clique-Width at most 4 and that a Clique-Width expression
can be computed in linear time.
