Local Boxicity, Local Dimension, and Maximum Degree

In this paper, we focus on two recently introduced parameters in the literature, namely `local boxicity' (a parameter on graphs) and `local dimension' (a parameter on partially ordered sets). We give an `almost linear' upper bound for both the parameters in terms of the maximum degree of a graph (for local dimension we consider the comparability graph of a poset). Further, we give an $O(n\Delta^2)$ time deterministic algorithm to compute a local box representation of dimension at most $3\Delta$ for a claw-free graph, where $n$ and $\Delta$ denote the number of vertices and the maximum degree, respectively, of the graph under consideration. We also prove two other upper bounds for the local boxicity of a graph, one in terms of the number of vertices and the other in terms of the number of edges. Finally, we show that the local boxicity of a graph is upper bounded by its `product dimension'.Comment: 11 page

Revisiting Interval Graphs for Network Science

The vertices of an interval graph represent intervals over a real line where overlapping intervals denote that their corresponding vertices are adjacent. This implies that the vertices are measurable by a metric and there exists a linear structure in the system. The generalization is an embedding of a graph onto a multi-dimensional Euclidean space and it was used by scientists to study the multi-relational complexity of ecology. However the research went out of fashion in the 1980s and was not revisited when Network Science recently expressed interests with multi-relational networks known as multiplexes. This paper studies interval graphs from the perspective of Network Science

Boxicity and separation dimension

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.Comment: This is the full version of a paper by the same name submitted to WG-2014. Some results proved in this paper are also present in arXiv:1212.6756. arXiv admin note: substantial text overlap with arXiv:1212.675