"Bipartite Probe Interval Graphs, Interval Point Bigraphs, and Circular" by David E. Brown and J. R. Lundgren
 

Bipartite Probe Interval Graphs, Interval Point Bigraphs, and Circular ArcGraphs

Document Type

Article

Journal/Book Title/Conference

The Australasian Journal of Combinatorics

Volume

35

Publication Date

2006

First Page

221

Last Page

236

Abstract

An intersection graph is a graph whose vertices are in bijective correspondence to a collection of sets so that vertices are adjacent if and only if their corresponding sets intersect. A graph G is a probe interval graph if it has a vertex partition V(G)=(P,N) and an interval of ℝ assigned to each vertex such that vertices are adjacent if and only if their corresponding intervals intersect and at least one of the vertices belongs to P. The sets P and N are called the probes and nonprobes, respectively. A circular arc graph is the intersection graph of arcs of a circle. An interval point bigraph is a bipartite intersection graph of points and intervals, that is, a graph G with bipartition V(G)=X∪Y in which one of the partite sets corresponds to a collection of points of ℝ and the other to intervals with vertices adjacent if and only if the point for one is contained in the interval for the other. We show that the complements of a class of 2-clique circular arc graphs, and a class of bipartite probe interval graphs are each equivalent to interval point bigraphs. Specifically, we characterize the bipartite probe interval graphs in which the probe/nonprobe partition can correspond to the bipartition. We also give a characterization for the aforementioned via a consecutively orderable edge partition into stars, and a new characterization for probe interval graphs by a consecutively orderable collection of quasi cliques.

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