Even and odd pairs in linegraphs of bipartite graphs

Two vertices in a graph are called an even pair (odd pair) if all induced paths between these two vertices have even (odd) length. Even and odd pairs have turned out to be of importance in conjunction with perfect graphs. We will characterize all linegraphs of bipartite graphs that contain an even resp. odd pair. In general, it is a co-NP-complete problem to decide whether a graph contains an even pair. For the class of linegraphs of bipartite graphs we will show that testing for even resp. odd pairs can be done in polynomial time.