Linear-Time Recognition of Probe Interval Graphs

The interval graph for a set of intervals on a line consists of one vertex for each interval, and an edge for each intersecting pair of intervals. A probe interval graph is a variant that is motivated by an application to genomics, where the intervals are partitioned into two sets: probes and non-probes. The graph has an edge between two vertices if they intersect and at least one of them is a probe. We give a linear-time algorithm for determining whether a given graph and partition of vertices into probes and non-probes is a probe interval graph. If it is, we give a layout of intervals that proves this. We can also determine whether the layout of the intervals is uniquely constrained within the same time bound. As part of the algorithm, we solve the consecutive-ones probe matrix problem in linear time, develop algorithms for operating on PQ trees, and give results that relate PQ trees for different submatrices of a consecutive-ones matrix.