A tree representation for P4-sparse graphs

Abstract

A graph G is P 4 -sparse if no set of five vertices in G induces more than one chordless path of length three. P 4 -sparse graphs generalize both the class of cographs and the class of P 4 -reducible graphs. We give several characterizations for P 4 -sparse graphs and show that they can be constructed from single-vertex graphs by a finite sequence of operations. Our characterization implies that the P 4 -sparse graphs admit a tree representation unique up to isomorphism. Furthermore, this tree representation can be obtained in polynomial time.