Bipartite Probe Interval Graphs, Interval Point Bigraphs, and Circular ArcGraphs
The Australasian Journal of Combinatorics
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.
Brown, D. E and J. R. Lundgren, Bipartite Probe Interval Graphs, Interval Point Bigraphs, and Circular Arc Graphs, Australasian Journal of Combinatorics, Volume 35, pp. 221 - 236, 2006.