Linear time recognition algorithms and structure theorems for bipartite tolerance graphs and bipartite probe interval graphs

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...

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Bibliographic Details
Published inDiscrete Mathematics and Theoretical Computer Science Vol. 12; no. 5; p. 63
Main Authors Brown, David E, Busch, Arthur H, Isaak, Garth
Format Journal Article
LanguageEnglish
Published DMTCS 01.01.2011
Subjects
Algorithms
Graphic methods
Investigations
Theorems (Mathematics)
United States
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Summary: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.sub.v] of the real line and a positive real number [t.sub.v] such that uv ∈ E if and only if [absolute value of [I.sub.u] ∩ [I.sub.v] [I.sub.v]] ≥ min{[t.sub.u], [t.sub.v]}. In this paper we present O([absolute value of V] + [absolute value of 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. Keywords: tolerance graphs, probe interval graphs, recognition algorithms
ISSN:1462-7264