Abstract
Two edges are called P 4-adjacent if they belong to the same P 4 (chordless path on four 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 orienting P 4-comparability graphs and of recognizing P 4-indifference graphs from O( n 5) and O( n 6) to O( m 2). 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 (SIAM J. Discrete Math. 8 (1995) 448–463).
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