NC Algorithms for Comparability Graphs, Interval Graphs, and Unique Perfect Matchings

Abstract

Laszlo Lovasz recently posed the following problem: "Is there an NC algorithm for testing if a given graph has a unique perfect matching?" We present such an algorithm for bipartite graphs. We also give NC algorithms for obtaining a transitive orientation of a comparability graph, and an interval representation of an interval graph. These enable us to obtain an NC algorithm for finding a maximum matching in an incomparability graph.