Two edges are called P4-adjacent if they belong to the same P4 (chordless path on four vertices). P4-components, in our terminology, are the equivalence classes of the transitive closure of the P4-adjacency relation. In this paper, new results on the structure of P4-components are obtained. On the one hand, these results allow us to improve the complexity of orienting P4-comparability graphs and of recognizing P4-indifference graphs from O(n5) and O(n6) to O(m2). On the other hand, by combining the modular decomposition with the substitution of P4-components, a new unique tree representation for arbitrary graphs is derived which generalizes the homogeneous decomposition introduced by Jamison and Olariu.
