dc.contributor.author |
Gangopadhyay, Rahul |
|
dc.contributor.author |
Shannigrahi, Saswata (Advisor) |
|
dc.contributor.author |
Sharma, Anuradha (Advisor) |
|
dc.date.accessioned |
2020-09-15T11:06:18Z |
|
dc.date.available |
2020-09-15T11:06:18Z |
|
dc.date.issued |
2020-01 |
|
dc.identifier.uri |
http://repository.iiitd.edu.in/xmlui/handle/123456789/828 |
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dc.description.abstract |
In this thesis, we study the d-dimensional rectilinear drawings of d-uniform hypergraphs in which each hyperedge contains exactly d vertices. A d-dimensional rectilinear drawing of a d-uniform hypergraph is a drawing of the hypergraph in Rd when its vertices are placed as points in general position and its hyperedges are drawn as the convex hulls of the corresponding d points. In such a drawing, a pair of hyperedges forms a crossing if they are vertex disjoint and contain a common point in their relative interiors. A special kind of d-dimensional rectilinear drawing of a d-uniform hypergraph is known as a d-dimensional convex drawing of it when its vertices are placed as points in general as well as in convex position in Rd. The d-dimensional rectilinear crossing number of a d-uniform hypergraph is the minimum number of crossing pairs of hyperedges among all d-dimensional rectilinear drawings of it. Similarly, the d-dimensional convex crossing number of a d-uniform hypergraph is the minimum number of crossing pairs of hyperedges among all d-dimensional convex drawings of it.
We study two types of uniform hypergraphs in this thesis, namely, the complete d-uniform hypergraphs and the complete balanced d-partite d-uniform hypergraphs. We summarise the main results of this thesis as follows.
We prove that the d-dimensional rectilinear crossing number of a complete d-uniform hypergraph having n vertices is Ω(2^d⋅√d)(█(n@2d)).
We prove that any 3-dimensional convex drawing of a complete 3-uniform hypergraph with n vertices contains 3(n¦6) crossing pairs of hyperedges.
We prove that there exist θ(4^d/√d)(█(n@2d)) crossing pairs of hyperedges in the d-dimensional rectilinear drawing of a complete d-uniform hypergraph having n vertices when all its vertices are placed over the d-dimensional moment curve.
We prove that the d-dimensional rectilinear crossing number of a complete balanced d-partite d-uniform hypergraph having nd vertices is Ω(2^d ) (n/2)^d (((n-1))/2)^d.
We also study the properties of different types of d-dimensional rectilinear drawings and d-dimensional convex drawings of the complete d-uniform hypergraph having 2d vertices by exploiting its relations with convex polytopes and k-sets. |
en_US |
dc.language.iso |
en |
en_US |
dc.publisher |
IIIT-Delhi |
en_US |
dc.subject |
CSE | Gale Transformation | d-Uniform Hypergraphs | Ham-Sandwich |
en_US |
dc.title |
Rectilinear crossing number of uniform hypergraphs |
en_US |
dc.type |
Thesis |
en_US |