Metric graphs elastically embeddable in the plane

Abstract

We study weighted graphs that can be embedded in the plane in such a way as to preserve an edge's weight as Euclidean distance between its two endpoints. Such questions arise in a variety of layout problems. In automatic graph drawing, for example, vertices are to be placed so as to approximate desired pairwise distances. The analogous 3-d problem arises in the distance geometry approach to molecular modeling, where edge weights are approximate distance measurements. We introduce the concept of elastic embeddability designed to deal with distances subject to error. Elastic graphs are related to, but distinct from, generically rigid graphs known in structural engineering. As an example that can be proven from basic graph-theoretic concepts, we characterize a subclass of elastic graphs: A chordal biconnected graph is elastically embeddable in the plane iff it does not contain the complete graph K4 as a subgraph.