Realizability of graphs

Abstract

A graph is d-realizable if, for every configuration of its vertices in EN, there exists a another corresponding configuration in E d 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. © 2007 Springer Science + Business Media, Inc.