Ferrers Dimension and Boxicity

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

💡 Research Summary

The paper investigates the relationship between two graph‑theoretic parameters that arise from different representations: the boxicity of undirected graphs and the Ferrers dimension of directed graphs. A Ferrers digraph is defined as a digraph whose out‑neighbour sets are linearly ordered by inclusion; equivalently, its adjacency matrix can be permuted so that all 1‑entries occupy a Ferrers diagram (a corner of the matrix). Every digraph can be expressed as the intersection of a finite family of Ferrers digraphs, and the minimum number of such digraphs is called the Ferrers dimension dF(D).

Boxicity, introduced by Roberts, measures how many dimensions of axis‑parallel boxes are needed to represent a graph as an intersection graph. A graph has boxicity 0 iff it is complete, and boxicity 1 exactly characterises interval graphs.

The central result (Theorem 2.1) shows that for any undirected graph G with a loop at every vertex, there exists a digraph D such that G = D ∩ Dᵀ and box(G) = dF(D). The proof proceeds by taking a box representation of G with n dimensions, converting each dimension’s interval graph Iᵢ into a Ferrers digraph Fᵢ (using Observation 1.1), and observing that D = ⋂₁ⁿ Fᵢ satisfies dF(D) ≤ n. A contradiction argument shows that dF(D) cannot be smaller than n, establishing equality. Consequently, for any D with G = D ∩ Dᵀ we always have box(G) ≤ dF(D), and the minimum of dF(D) over all such D equals box(G).

An analogous correspondence is proved for bipartite graphs. For a bipartite graph B, define bB as the graph obtained by making each partite set a clique and adding loops at all vertices. Theorem 2.2 establishes dF(B) = box(bB). This yields a clean characterisation: B has Ferrers dimension at most 2 iff bB is a 2‑clique rectangular graph (the intersection graph of axis‑parallel rectangles whose vertices can be covered by two disjoint cliques). Moreover, B is an interval bipartite graph iff bB is a 2‑clique rectangular graph admitting a rectangle representation where every pair of rectangles intersect on at least one coordinate axis (Corollary 2.4).

The paper further explores quantitative bounds linking boxicity and Ferrers dimension. For a symmetric digraph D(G) obtained from an undirected graph G (i.e., D(G) shares G’s adjacency matrix), let k = dF(D(G)). Theorem 2.5 proves k² ≤ box(G) ≤ k – 1, and both bounds are tight. The lower bound is attained by the 4‑cycle C₄ (box(C₄)=2, dF(D(C₄))=4), while the upper bound is attained by the 6‑cycle C₆ (box(C₆)=2, dF(D(C₆))=3). The proof uses the fact that each Ferrers component contributes at most two interval graphs to the box representation, and that a Ferrers digraph with loops yields a complete bipartite subgraph in the associated auxiliary graph H(D).

Overall, the work bridges the gap between geometric intersection representations of undirected graphs and combinatorial matrix‑permutation characterisations of directed graphs. By translating a box representation into an intersection of Ferrers digraphs (and vice‑versa), the authors provide a new tool for estimating or computing boxicity via Ferrers dimension, and they obtain new characterisations of bipartite graphs of low Ferrers dimension. These results have potential applications in scheduling, resource allocation, and any domain where graph representations by intervals or boxes are used, offering a unified perspective that connects interval/box models with Ferrers matrix structures.