This paper investigates ultrafilters in the context of connectivity systems, defined as pairs
$(X, f) $ where $ X $ is a finite set and $ f $ is a symmetric submodular function. Ultrafilters …

The boxicity of a graph is the smallest dimension d allowing a representation of it as the
intersection graph of a set of d-dimensional axis-parallel boxes. We present a simple …

Let $ box (G) $ be the boxicity of a graph $ G $, $ G [H_1, H_2,\ldots, H_n] $ be the $ G $-
generalized join graph of $ n $-pairwise disjoint graphs $ H_1, H_2,\ldots, H_n $, $ G^ d_k …

The boxicity (cubicity) of a graph G is the minimum natural number k such that G can be
represented as an intersection graph of axis-parallel rectangular boxes (axis-parallel unit …

The separation dimension of a graph G is the smallest natural number k for which the
vertices of G can be embedded in R^k such that any pair of disjoint edges in G can be …

A family 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 F in which all the vertices …

The boxicity of a graph is the smallest dimension $ d $ allowing a representation of it as the
intersection graph of a set of $ d $-dimensional axis-parallel boxes. We present a simple …

Abstract A k-box B=(R 1,…, R k), where each R i is a closed interval on the real line, is
defined to be the Cartesian product R 1× R 2×⋯× R k. If each R i is a unit-length interval, we …

Planar bipartite graphs can be represented as touching graphs of horizontal and vertical
segments in R^ 2 R 2. We study a generalization in space, namely, touching graphs of axis …

The separation dimension of a graph G is the minimum positive integer d for which there is
an embedding of G into ℝd, such that every pair of disjoint edges are separated by some …