Ferrers Dimension Research Articles

Sublinear approximation algorithms for boxicity and related problems

The boxicity of a graph G(V,E) is the minimum integer k such that G can be represented as the intersection graph of axis parallel boxes in Rk. Cubicity is a variant of boxicity, where the axis parallel boxes in the intersection representation are restricted to be of unit length sides. Deciding whether the boxicity (resp. cubicity) of a graph is at most k is NP-hard, even for k=2 or 3. Computing these parameters is inapproximable within O(n1−ϵ)-factor, for any ϵ>0 in polynomial time unless NP=ZPP, even for many simple graph classes.In this paper, we give a polynomial time κ(n) factor approximation algorithm for computing boxicity and a κ(n)⌈loglogn⌉ factor approximation algorithm for computing the cubicity, where κ(n)=2nloglogn∕logn. These o(n) factor approximation algorithms also produce the corresponding box (resp. cube) representations. As a special case, this resolves the question posed by Spinrad (2003) about polynomial time construction of o(n) dimensional box representations for boxicity 2 graphs. Other consequences of our approximation algorithm include O(κ(n)) factor approximation algorithms for computing the following parameters: the partial order dimension (poset dimension) of finite posets, the interval dimension of finite posets, minimum chain cover of bipartite graphs, Ferrers dimension of digraphs and threshold dimension of split graphs and co-bipartite graphs. Each of these parameters is inapproximable within an O(n1−ϵ)-factor, for any ϵ>0 in polynomial time unless NP=ZPP and the algorithms we derive seem to be the first o(n) factor approximation algorithms known for all these problems. We note that obtaining a o(n) factor approximation for poset dimension was also mentioned as an open problem by Felsner et al. (2017).In the second part of this paper, parameterized approximation algorithms for boxicity using various edit distance parameters are derived. We also present a parameterized approximation scheme for cubicity, using minimum vertex cover number as the parameter.

Ferrers dimension of grid intersection graphs

We investigate the Ferrers dimension of classes of grid intersection graphs and show properties and characterizations. In particular, we show that (1) the grid intersection graphs form a proper subclass of the class of bipartite graphs of Ferrers dimension 4, (2) segment-ray graphs have a forbidden submatrix characterization, and (3) a bipartite graph is a unit grid intersection graph if and only if it is the intersection of two bipartite permutation graphs.

Adjacency matrices of probe interval graphs

In this paper we obtain several characterizations of the adjacency matrix of a probe interval graph. In course of this study we describe an easy method of obtaining interval representation of an interval bigraph from its adjacency matrix. Finally, we note that if we add a loop at every probe vertex of a probe interval graph, then the Ferrers dimension of the corresponding symmetric bipartite graph is at most 3.

Ferrers dimension and boxicity

This note explores the relation between the boxicity of undirected graphs and the Ferrers dimension of directed graphs.

Bigraphs/digraphs of Ferrers dimension 2 and asteroidal triple of edges

Three mutually separable edges of a graph form an asteroidal triple of edges (ATE), if for any two of them, there is a path from the vertex set of one to the vertex set of another that avoids the neighbours of the third edge. In this note, it is shown that a bigraph of Ferrers dimension at most 2 is bichordal and ATE-free, but the converse is not true.

Intersection representation of digraphs in trees with few leaves

The leafage of a digraph is the minimum number of leaves in a host tree in which it has a subtree intersection representation. We discuss bounds on the leafage in terms of other parameters (including Ferrers dimension), obtaining a string of sharp inequalities. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 340–353, 1999

Representing digraphs using intervals or circular arcs

Containment and overlap representations of digraphs are studied, with the following results. The interval containment digraphs are the digraphs of Ferrers dimension 2, and the circular-arc containment digraphs are the complements of circular-arc intersection digraphs. A poset is an interval containment poset if and only if its comparability digraph is an interval containment digraph, and a graph is an interval graph if and only if the corresponding symmetric digraph with loops is an interval digraph. In an appropriate model of overlap representation using intervals, the unit right-overlap interval digraphs are precisely the unit interval digraphs, and the adjacency matrices of right-overlap interval digraphs have a simple structural characterization bounding their Ferrers dimension by 3.

On the 2-Chain Subgraph Cover and Related Problems

In this paper, we provide efficient algorithms for a number of closely related problems. They are: threshold number 2 on split graphs, Ferrers dimension 2, biorder dimension 2, interval dimension 2, and the trapezoid graph recognition problems. We show that all these problems can be reduced to the 2-chain subgraph cover problem in O(n2) time. By introducing a new algorithm to reduce a 2-chain subgraph cover problem to a partial order dimension 2 problem in O(n2) time, we find an O(n2) time algorithm to solve the 2-chain subgraph cover problem. As a consequence, all the previously mentioned problems can be solved in O(n2) time. No other algorithm known to the authors can solve any of these problems in the same time bound.

An interval digraph in relation to its associated bipartite graph

The intersection digraph of a family of ordered pairs of sets {( S v , T v ): v∈ V} is the digraph D( V, E) such that uv∈ E if and only if S u ∩ T v ≠0. Internal digraphs are those intersection digraphs for which the subsets are intervals on the real line. In a previous paper, they were characterized in terms of Ferrers digraphs and a close relationship was obtained between an interval digraph and a digraph of Ferrers dimension 2. In order to characterize a digraph D of Ferrers dimension 2, Cogis associated an undirected graph H( D) with D in a suitable way, the vertices of H( D) corresponding to the zeros of the adjacency matrix of D. He proved that D has Ferrers dimension at most 2 if and only if H( D) is bipartite. Depending on the above characterization, this paper first obtains some properties of a digraph of Ferrers dimension 2; then it is shown how the notion of interior edges is related to an interval digraph.

The order dimension of multinomial lattices

Using the notion of Ferrers dimension of incidence structures, the order dimension of multi-nomial lattices (i.e. lattices of ‘multi-permutations’) is determined. In particular, it is shown that the lattice of all permutations on ann-element set has dimensionn−1.

Removing critical pairs

A long-standing conjecture in dimension theory asserts that every ordered set with at least three elements contains two elements whose removal decreases the dimension by at most one. We disprove the yet stronger conjecture that the removal of any critical pair decreases the dimension by at most one. We have constructed the counterexample using Ferrers dimension which will be described in this paper.

On the Ferrers dimension of a digraph

The Ferrers dimension of a digraph has been shown to be an extension of the order dimension. By proving a property of (finite) transitive Ferrers digraphs, we give an original proof of this above result and derive Ore's alternative definition of the order dimension. Still, the order dimension is proved to be ‘polynomially equivalent’ to the Ferrers dimension.