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The realizability of a graph is the smallest dimension, d, in which for any realization (placement of the vertices) of the graph in any N-dimensional Euclidean space, there exists a realization of the graph with the same edge lengths in d-dimensional Euclidean space. Expanding on the work of Belk and Connelly who determined the set of all forbidden minors for dimensions up to 3, we determine a large family of forbidden minors for each dimension greater than 3. At the heart of this graph family is a new concept, spherical realizability, which places the vertices of a graph on a d-sphere, rather than in Euclidean space. In addition, we prove theorems regarding rigidity and realizability, and we bound above and below the realizability of certain graph families including bipartite graphs, powers of cycles, and complements of powers of cycles.
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Winslow, Noah Schuyler, "On the Realizability and Rigidity of Graphs in Higher Dimensions" (2015). Senior Projects Spring 2015. 368.