Recognizing $P_4 $-Sparse Graphs in Linear Time

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. One remarkable feature of $P_4 $-sparse graphs is that they admit a tree representation unique up to isomorphism. It has been shown that this tree representation can be obtained in polynomial time. This paper gives a linear time algorithm to recognize $P_4 $-sparse graphs and shows how the data structures returned by the recognition algorithm can be used to construct the corresponding tree representation in linear time.