Directed tolerance graphs

In 1969, Tucker characterized proper circular-arc graphs as those graphs whose augmented adjacency matrices have the circularly compatible ones property. Moreover, he also found a polynomial-time a...

Circularly Compatible Ones, $D$-Circularity, and Proper Circular-Arc Bigraphs

An interval bigraph B is a proper interval bigraph if there is an interval representation of B such that no interval of the same partite set is properly contained in the other. Similarly a circular arc bigraph B is a proper circular arc bigraph if there is a circular arc representation of B such that no arc of the same partite set is properly contained in the other. In this paper, we characterize proper interval bigraphs and proper circular arc bigraphs using two linear orderings of their vertex set.

New Characterizations of Proper Interval Bigraphs and Proper Circular Arc Bigraphs

Graphs and digraphs represented by intervals and circular arcs

We consider a special type of interval catch digraph (ICD) family for one-dimensional data in a randomized setting and propose its use for testing uniformity. These ICDs are defined with an expansion and a centrality parameter, hence we will refer to this ICD as parameterized ICD (PICD). We derive the exact (and asymptotic) distribution of the domination number of this PICD family when its vertices are from a uniform (and non-uniform) distribution in one dimension for the entire range of the parameters; thereby determine the parameters for which the asymptotic distribution is non-degenerate. We observe jumps (from degeneracy to non-degeneracy or from a non-degenerate distribution to another) in the asymptotic distribution of the domination number at certain parameter combinations. We use the domination number for testing uniformity of data in real line, prove its consistency against certain alternatives, and compare it with two commonly used tests and three recently proposed tests in literature and also arc density of this ICD and of another ICD family in terms of size and power. Based on our extensive Monte Carlo simulations, we demonstrate that domination number of our PICD has higher power for certain types of deviations from uniformity compared to other tests.

Domination Number of an Interval Catch Digraph Family and its use for Testing Uniformity

Forbidden substructure for interval digraphs/bigraphs

New characterizations of proper interval bigraphs

A bipartite graph B with bipartion X, Y is called a Ferrers bigraph if the neighbor sets of the vertices of X (or equivalently Y) are linearly ordered by set inclusion. The Ferrers dimension of B is the minimum number of Ferrers bigraphs whose intersection is B. In this paper we present a new approach of finding the forbidden subgraphs of bigraphs of Ferrers dimension 2 when it contains a strong bisimplicial edge.

Ashok Kumar Das

Papers

Forbidden substructure for interval digraphs/bigraphs

New characterizations of proper interval bigraphs

New Characterizations of Proper Interval Bigraphs and Proper Circular Arc Bigraphs

Graphs and digraphs represented by intervals and circular arcs

Forbidden Subgraphs of Bigraphs of Ferrers Dimension 2