### Linear Time Recognition Algorithms and Structure Theorems for Bipartite Tolerance Graphs and Bipartite Probe Interval Graphs

*David E. Brown, Arthur H. Busch, Garth Isaak*

#### Abstract

A graph is a

*probe interval graph*if its vertices can be partitioned into probes and nonprobes with an interval associated to each vertex so that vertices are adjacent if and only if their corresponding intervals intersect and at least one of them is a probe. A graph G = (V,E) is a tolerance graph if each vertex v ∈V can be associated to an interval I_{v}of the real line and a positive real number t_{v}such that uv ∈E if and only if |I_{u}∩I_{v}| ≥ min (t_{u},t_{v}). In this paper we present O(|V| + |E|) recognition algorithms for both bipartite probe interval graphs and bipartite tolerance graphs. We also give a new structural characterization for each class which follows from the algorithms.Full Text: PDF PostScript