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Realizability of Graphs and Linkages

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Thirty Essays on Geometric Graph Theory

Abstract

We show that deciding whether a graph with given edge lengths can be realized by a straight-line drawing has the same complexity as deciding the truth of sentences in the existential theory of the real numbers, ETR; we introduce the class \(\exists \mathbb{R}\) that captures the computational complexity of ETR and many other problems. The graph realizability problem remains \(\exists \mathbb{R}\)-complete if all edges have unit length, which implies that recognizing unit distance graphs is \(\exists \mathbb{R}\)-complete. We also consider the problem for linkages: In a realization of a linkage, vertices are allowed to overlap and lie on the interior of edges. Linkage realizability is \(\exists \mathbb{R}\)-complete and remains so if all edges have unit length. A linkage is called rigid if any slight perturbation of its vertices that does not break the linkage (i.e., keeps edge lengths the same) is the result of a rigid motion of the plane. Testing whether a configuration is not rigid is \(\exists \mathbb{R}\)-complete.

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Notes

  1. 1.

    A manuscript collecting many of these problems is in preparation (Schaefer, The real logic of drawing graphs, personal communication).

  2. 2.

    When writing formulas in the existential theory of the reals, we will freely use integers and rationals, since these can easily be eliminated without affecting the length of the formula substantially. We will also drop the symbol ∗ .

  3. 3.

    Tarski showed that the full theory of the reals is decidable by quantifier elimination.

  4. 4.

    This reducibility is known as polynomial-time many–one reducibility; since we have no need for other reducibilities in this chapter, we simplify to “reduces.” We consider decision problems (requiring a “yes” or “no” answer) encoded as sets of binary strings; that is, \(A,B \subseteq \{ 0, {1\}}^{{_\ast}}\), and \(f :\{ 0, {1\}}^{{_\ast}}\rightarrow \{ 0, {1\}}^{{_\ast}}\). For more background on encodings and basic definitions from computational complexity including complexity classes NP (nondeterministic polynomial time) and PSPACE (polynomial space), see any of the standard references, e.g., [35, 44]. We could have replaced polynomial time with logarithmic space in the definition of ≤ m but decided to use the more familiar notion.

  5. 5.

    So “A reduces to B” does not mean that B is easier than A, as the word “reduces” may incorrectly suggest.

  6. 6.

    The statement in [2] contains a typo in the radius of the ball.

  7. 7.

    For a discussion of graph drawing assumptions, see [47].

  8. 8.

    Our distinction between the realizability of graphs and linkages is not universal, but not unusual; for linkages, [10] is a good reference; for graph realizability, definitions and terminology vary. For instance, realizable graphs are sometimes called Euclidean graphs. Often the definitions allow that vertices lie on edges of which they are not endpoints, though that may in some cases be due to oversight; as we will see, from a computational point of view, this distinction does not matter.

  9. 9.

    There are many applets implementing Peaucellier’s linkage available on the web to play with [30]. James Joseph Sylvester writes about Lord Kelvin that he “nursed it as if it had been his own child, and when a motion was made to relieve him of it, replied ‘No! I have not had nearly enough of it—it is the most beautiful thing I have ever seen in my life’.” [46]. There have been other devices, both earlier and later, achieving the same effect, see [26] for an early history.

  10. 10.

    Erdős et al.  [16] defined the dimension of a graph using (noninduced) subgraphs of E n. Later, Erdős and Simonovits [15] introduced Euclidean dimension under the name faithful dimension. The two notions differ: Take a wheel W 6 with six spokes and remove one of the spokes. The resulting graph is realizable as a subgraph, but not as an induced subgraph of E 2. The name “Euclidean dimension” seems to be due to Maehara [31]. For details and more terminology and history, see [45, Sect. 13.2].

  11. 11.

    See [25] or [11, Sect. 3.2.1] for detailed expositions.

  12. 12.

    Kempe used a parallelism gadget like this in his proof of the universality theorem that every bounded part of an algebraic curve can be traced by a suitable linkage. His parallelism gadget was flawed, however; there are many ways to repair the construction. We follow a construction due to Kapovich and Millson [25], also described in [11, Sect. 3.2.2].

  13. 13.

    The realization of the Moser graph is not unique; both of the diamonds can flip; to force d and e to be realized at distance < 1 ∕ 2 as shown in Fig. 4, we brace the construction by adding edges gx, xy, and yb of lengths w(gx) = 1, w(xy) = 3, w(yb) = 1; this forces g and b to have distance at least 1, thereby forcing the intended realization of the Moser graph.

  14. 14.

    The name H 2 N seems to be short for Hilbert’s homogeneous Nullstellensatz [27].

  15. 15.

    This is a folklore result; for example, it is easy to see that STRETCHABILITY can be rephrased like this. The version in the Blum–Shub–Smale model can be found in [6].

  16. 16.

    The reduction is polynomial time, since our polynomials have total degree 2.

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Acknowledgements

I would like to thank the anonymous referee for pointing out the universality papers by Jordan and Steiner [23] and Kapovich and Millson [25].

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  1. Department of Computer Science, DePaul University, 243 South Wabash, Chicago, Illinois, 60604, USA

    Marcus Schaefer

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  1. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Hungary

    János Pach (Chair of Combinatorial Geometry) (Chair of Combinatorial Geometry)

  2. École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland

    János Pach (Chair of Combinatorial Geometry) (Chair of Combinatorial Geometry)

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Schaefer, M. (2013). Realizability of Graphs and Linkages. In: Pach, J. (eds) Thirty Essays on Geometric Graph Theory. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0110-0_24

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