131 research outputs found
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 time deterministic algorithm to compute a local box
representation of dimension at most for a claw-free graph, where
and 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
Increasing United States Investment in Foreign Securities: An Evaluation of SEC Rule 144A
The dimension of a poset is the minimum number of total orders whose intersection is . We prove that the dimension of every poset whose comparability graph has maximum degree is at most . This result improves on a 30-year old bound of Füredi and Kahn and is within a factor of optimal. We prove this result via the notion of boxicity. The boxicity of a graph is the minimum integer such that is the intersection graph of -dimensional axis-aligned boxes. We prove that every graph with maximum degree has boxicity at most , which is also within a factor of optimal. We also show that the maximum boxicity of graphs with Euler genus is , which solves an open problem of Esperet and Joret and is tight up to a constant factor
On (2,3)-agreeable Box Societies
The notion of -agreeable society was introduced by Deborah Berg et
al.: a family of convex subsets of is called -agreeable if any
subfamily of size contains at least one non-empty -fold intersection. In
that paper, the -agreeability of a convex family was shown to imply the
existence of a subfamily of size with non-empty intersection, where
is the size of the original family and is an explicit
constant depending only on and . The quantity is called
the minimal \emph{agreement proportion} for a -agreeable family in
.
If we only assume that the sets are convex, simple examples show that
for -agreeable families in where . In this paper,
we introduce new techniques to find positive lower bounds when restricting our
attention to families of -boxes, i.e. cuboids with sides parallel to the
coordinates hyperplanes. We derive explicit formulas for the first non-trivial
case: the case of -agreeable families of -boxes with .Comment: 15 pages, 10 figure
Boxicity and separation dimension
A family of permutations of the vertices of a hypergraph is
called 'pairwise suitable' for if, for every pair of disjoint edges in ,
there exists a permutation in in which all the vertices in one
edge precede those in the other. The cardinality of a smallest such family of
permutations for is called the 'separation dimension' of and is denoted
by . Equivalently, is the smallest natural number so that
the vertices of can be embedded in such that any two
disjoint edges of can be separated by a hyperplane normal to one of the
axes. We show that the separation dimension of a hypergraph is equal to the
'boxicity' of the line graph of . 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
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
- …