Linear structure of bipartite permutation graphs and the longest path problem

Abstract

The class of bipartite permutation graphs is the intersection of two well known graph classes: bipartite graphs and permutation graphs. A complete bipartite decomposition of a bipartite permutation graph is proposed in this note. The decomposition gives a linear structure of bipartite permutation graphs, and it can be obtained in O ( n ) time, where n is the number of vertices. As an application of the decomposition, we show an O ( n ) time and space algorithm for finding a longest path in a bipartite permutation graph.