Planar Embeddings of Graphs with Specified Edge Lengths

Abstract

Abstract.We consider the problem of finding a planar embedding of a (planar) graph with a prescribed Euclidean length on every edge. There has been substantial previous work on the problem without the planarity restrictions, which has close connections to rigidity theory, and where it is easy to see that the problem is NP-hard. In contrast, we show that the problem is tractable-indeed, solvable in linear time on a real RAM-for planar embeddings of planar 3-connected triangulations, even if the outer face is not a triangle. This result is essentially tight: the problem becomes NP-hard if we consider instead planar embeddings of planar 3-connected infinitesimally rigid graphs, a natural relaxation of triangulations in this context.

Realizability of Graphs and Linkages

“…Strengthening earlier results by Whitesides [49] and Eades and Wormald [13], Cabello, Demaine, and Rote [7] showed that the plane realizability problem is (strongly) NP-hard even when restricted to 3-connected, infinitesimally rigid planar graphs with unit edge lengths. Plane realizable graphs with unit edge lengths-plane unit distance graphs-are known as matchstick graphs.…”

“…Realizability depends on the notion of drawing we use; in the standard definition of a drawing, different vertices cannot coincide in the drawing and a vertex cannot lie on an edge unless it is an endpoint of that edge. 7 If we do allow vertices to coincide and lie on edges, we enter the realm of linkages. For example, K 2,3 cannot be realized by a standard straight-line drawing in the plane if all edges have unit length, but it can be realized as a linkage with vertices overlapping.…”

Travel-Time Maps: Linear Cartograms with Fixed Vertex Locations

“…In addition to the results on linear cartograms [3,5,12,18], several other papers are also related. Weights of edges (but for very different applications) can also be visualized by the width [2].…”

“…"Relative lengths" of stretches are immediately quantified, whereas "relative colors" do not have a clear interpretation. These observations have led to the development of linear cartograms [3,5,12,18]. Linear cartograms try to display time more clearly by distorting the base map.…”

Time-Space Maps from Triangulations

“…For example, metro maps show connectivity of metro lines while abstracting from geographic reality (correct location) [11], and cartograms may show countries by using size (area) to depict total population [15]. Cartograms come in different types: contiguous area [7,9], non-contiguous area [12], rectangular [10], rectilinear [4], circle [6], and linear cartograms [5]. Except for the linear cartogram, all types use the area to show a variable of interest (often population).…”