Boxicity and Separation Dimension

Abstract

A family $\mathcal {F}$ of permutations of the vertices of a hypergraph $H$ is called pairwise suitable for $H$ if, for every pair of disjoint edges in $H$, there exists a permutation in $\mathcal {F}$ in which all the vertices in one edge precede those in the other. The cardinality of a smallest such family of permutations for $H$ is called the separation dimension of $H$ and is denoted by $\pi (H)$. Equivalently, $\pi (H)$ is the smallest natural number $k$ so that the vertices of $H$ can be embedded in $\mathbb {R}^k$ such that any two disjoint edges of $H$ can be separated by a hyperplane normal to one of the axes. We show that the separation dimension of a hypergraph $H$ is equal to the boxicity of the line graph of $H$. This connection helps us in borrowing results and techniques from the extensive literature on boxicity to study the concept of separation dimension.

Manu Basavaraju: Supported by the European Research Council under the European Union’s Seventh Framework Programme (FP/2007–2013)/ERC Grant Agreement n. 267959.

Rogers Mathew and Deepak Rajendraprasad: Supported by VATAT Postdoctoral Fellowship, Council of Higher Education, Israel.