## Theses, Dissertations, and Projects

2015

Honors Thesis

Computer Science

#### Keywords

Graph theory, Sparse matrices, Graph algorithms, Rigidity (Geometry), Sparse graphs, Sparsity, (3, 6) sparse graphs, Pebble games

#### Abstract

A skeletal framework is an embedding of a nite graph into d-sapce. A nite graph can be represented as G = (V;E), where V is a set of vertices and E a set of edges. The rigidity of a framework in d-dimensions can be analyzed by studying a combinatorial property of its graphical representation, namely its sparsity. A class of sparse graphs { (3; 6)-sparse graphs { describes the rigidity of frameworks in three dimensions. All frameworks in three-dimensions have a necessary condition for rigidity { their graphs must be maximally (3; 6)-sparse. However, this is not a su cient condition and not all maximally (3; 6)-sparse graphs represent a framework that is rigid in three-dimensions. The aim of this paper is to study the fundamental problems in (3; 6)-sparsity in relation to matroidal sparse graphs, a class of sparse graphs for which much work already exists in the literature. We propose a variant of the Pebble Game algorithm to answer the (3; 6) decision problem{ decide if a given graph is (3; 6)-sparse or not { and demonstrate why the Pebble Game algorithm cannot address the other problems in (3; 6)-sparsity. We also describe inductive methods for generating di erent classes of sparse graphs.

English