CSCE 496/896 Project Idea: Bipartite Interval Graphs

Due date for project write-ups: Dec. 11

An interval graph is constructed as follows. Given a set of intervals on the line, associate a vertex of the graph with each interval and join two vertices by an edge if the corresponding intervals overlap. An example of an interval graph is below. A through E are the intervals (and nodes), and an edge connects two nodes if the intervals overlap (the numeric values of the interval endpoints do not affect the graph in this example).

In the case of unique probes, every error-free hybridization matrix defines an interval graph on the vertex set of clones in which clones i and j are joined by an edge if they have a probe in common. Work by Benzer on fragments of bacteriophage T4 DNA led to the following problem: Given information about whether or not two fragments of a genome overlap, is the data consistent with the hypothesis that the genes are arranged in linear order? This is equivalent to the question whether the overlap graph is an interval graph.

A bipartite interval graph is an interval graph such that the set of vertices can be partitioned into two sets V and W such that each vertex in the graph is in V or W (but not both) and no edges exist between any pair of nodes in V and no edges exist between any pair of nodes in W. The figure above is a bipartite interval graph.

It is known in restriction mapping that if a DNA molecule is digested twice (by two restriction enzymes), the interval graph from resulting fragments is a bipartite interval graph [Waterman and Griggs, Bulletin of Mathematical Biology, 48:189-195, 1986].

PROJECT IDEAS:

1. Design an efficient algorithm to recognize a bipartite interval graph.

2. Identify and describe applications of bipartite interval graphs (and general interval graphs) to biological problems.