Abstract
A graph is d-realizable if, for every configuration of its vertices in EN, there exists a another corresponding configuration in Ed with the same edge lengths. A graph is 2-realizable if and only if it is a partial 2-tree, i.e., a subgraph of the 2-sum of triangles in the sense of graph theory. We show that a graph is 3-realizable if and only if it does not have K5 or the 1-skeleton of the octahedron as a minor.
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Belk, M., Connelly, R. Realizability of Graphs. Discrete Comput Geom 37, 125–137 (2007). https://doi.org/10.1007/s00454-006-1284-5
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DOI: https://doi.org/10.1007/s00454-006-1284-5