06 04 06 5 v 2 [ cs . D S ] 1 8 Se p 20 06 A new tractable combinatorial decomposition

This paper introduces the umodules, a generalisation of modules for the homogeneous relations. We first present some properties of the umodule family, then show that, if the homogeneous relation fulfills some natural axioms, the umodule family has a unique decomposition tree. We show that this tree can be computed in polynomial time, under a certain size assumption. We apply this theory to a new tournament decomposition and a graph decomposition. In both cases, the decomposition tree computing time becomes linear. Moreover, we characterise the completely decomposable tournaments. Finally we give polynomial-time algorithms for the maximal umodules, the undecomposability of a relation, and its decomposition tree. We conclude with further applications of homogeneous relations.