Linear time recognition of P4-indifference graphs

A P4 is a chordless path of four vertices. A graph is P4-indifference if it admits an ordering < on its vertex set such that every P4 abcd has a < b < c < d or d < c < b < a. Such an ordering is called a P4-indifference ordering. The P4-indifference graphs were introduced in [Chv84] as a particular class of perfectly orderable graphs. A graph is perfectly orderable if there exists an ordering on its vertex set for which the greedy colouring algorithm produces an optimal colouring. The first recognition algorithm for P4-indifference graphs is due to Hoàng and Reed and has the complexity of O(n6) [HR89]. They compute the equivalence classes of some relation on the P4’s of the graph. They then check that these classes do not contain a certain subgraph with 6 vertices. Later, Raschle and Simon, studying more carefully the P4’s relations, proposed an O(n2m) recognition algorithm [RS97]. Recently, Hoàng, Maffray and Noy gave a characterization by forbidden induced subgraphs [HMN99] and raised the question of the existence of a linear time recognition algorithm. We answer their question in the affirmative way using some of their theorems. Moreover our algorithm computes an adequate ordering of the vertices when it concludes that the input graph is P4-indifference.