On the P4-components of graphs

Two edges are called P 4-adjacent if they belong to the same P 4 (chordless path on 4 vertices). P 4-components, in our terminology, are the equivalence classes of the transitive closure of the P 4-adjacency relation. In this paper, new results on the structure of P 4-components are obtained. On the one hand, these results allow us to improve the complexity of the recognition and orientation algorithms for P 4-comparability and P 4-indiierence graphs from O(n 5) to O(n 2 m) and from O(n 6) to O(n 2 m), respectively. On the other hand, by combining the modular decomposition with the substitution of P 4-components, a new unique tree representation for arbitrary graphs is derived which generalizes the homogeneous decomposition introduced by Jamison and Olariu JO95].