Journal of Graph Algorithms and Applications the Complexity of Simultaneous Geometric Graph Embedding

Given a collection of planar graphs G1,. .. , G k on the same set V of n vertices, the simultaneous geometric embedding (with mapping) problem, or simply k-SGE, is to find a set P of n points in the plane and a bijection ϕ : V → P such that the induced straight-line drawings of G1,. .. , G k under ϕ are all plane. This problem is polynomial-time equivalent to weak rectilinear realiz-ability of abstract topological graphs, which Kynčl (doi:10.1007/s00454-010-9320-x) proved to be complete for ∃R, the existential theory of the reals. Hence the problem k-SGE is polynomial-time equivalent to several other problems in computational geometry, such as recognizing intersection graphs of line segments or finding the rectilinear crossing number of a graph. We give an elementary reduction from the pseudoline stretchability problem to k-SGE, with the property that both numbers k and n are linear in the number of pseudolines. This implies not only the ∃R-hardness result, but also a 2 2 Ω(n) lower bound on the minimum size of a grid on which any such simultaneous embedding can be drawn. This bound is tight. Hence there exists such collections of graphs that can be simultaneously embedded, but every simultaneous drawing requires an exponential number of bits per coordinates. The best value that can be extracted from Kynčl's proof is only 2 2 Ω(√ n) .


Introduction
Given graphs G 1 = (V, E 1 ), . . . , G k = (V, E k ) on n vertices, simultaneous geometric embedding (with mapping) or simply k-SGE, is the problem of finding a point set P ⊂ R 2 of size n and a bijection ϕ : V → P such that the induced straight-line drawings of G 1 , . . . , G k under ϕ are all plane [2]. The problem 1-SGE amounts to planarity testing. The problem 2-SGE is typically referred to simply as SGE. Fig. 1 shows an example of two graphs and a 2-SGE.
Early work on the topic focussed on the existence of k-SGEs for restricted graph classes. The authors of the original paper show that there is a pair of outerplanar graphs on the same vertex set that does not admit a 2-SGE. Similarly, they give a triple of paths that does not admit a 3-SGE. The authors also show that various other classes of graphs, such as a pair of caterpillars, an extended  star and a path, or two stars always admit a 2-SGE. The question of whether any two trees admit a 2-SGE remained open for six years, until the question was settled in the negative with a counterexample [3]. The most recent negative result gives a tree and a path that do not admit a 2-SGE [4]. Research has since focussed on other variations of the problem, such as simultaneous embedding with fixed edges (edges are drawn as arbitrary simple Jordan curves, but all graphs must use identical curves for identical edges) or matched drawings (vertices have fixed y-coordinates in all drawings, but may have different x-coordinates in each drawing). The decision problem 2-SGE was shown to be NP-hard [5].
The existential theory of the reals is the set of true sentences of the form ∃(x 1 , . . . , x n ) : ϕ(x 1 , . . . , x n ), where ϕ is a quantifier-free Boolean formula (without negation) over the signature (0, 1, +, * , <) interpreted over the universe of real numbers. The decision problem ETR asks whether a given sentence is true. The complexity class ∃R is defined as the set of decision problems that can be reduced to ETR in polynomial time. Hence a problem is ∃R-complete if it belongs to ∃R and every problem in ∃R can be reduced to it in polynomial time [6]. It is known that ∃R is included in PSPACE [7]. The class ∃R also contains NP, since Boolean satisfiability can be encoded as a decision problem on a set of polynomial inequalities.
In this paper, we prove that k-SGE is at least as hard as the pseudoline stretchability problem, which is known to be ∃R-complete. Together with the result of Estrella-Balderrama et al. [5] stating that k-SGE belongs to ∃R, this proves that k-SGE is ∃R-complete. The result follows from the fact that k-SGE is polynomially equivalent to weak rectilinear realizability of abstract topological graphs [8] and the fact that weak rectilinear realizability is ∃R-complete [1]. We give an alternative reduction from order type realizability.
Since k-SGE is ∃R-complete, it is polynomial-time equivalent to many other classical problems in computational geometry, such as finding the rectilinear crossing number of a graph [9], recognizing unit disk graphs [10], recognizing intersection graphs of convex sets in the plane [6], intersection graphs of segments [11], solving the Steinitz problem [12], and deciding the realizability of linkages [13]. We refer the reader to the recent works of Schaefer for more references and examples [6,14].
The result also implies that it is unlikely that there exists a polynomial-size description of any simultaneous geometric embedding. The proof indeed shows that for some positive instances of k-SGE, representing the point set by encoding the coordinates of each point requires an exponential number of bits. This follows from the analogous result on realizations of order types by Goodman, Pollack, and Sturmfels [15].
In Section 2, we briefly recall standard results on pseudoline arrangements and stretchability. The reduction itself is given in Section 3.

Pseudolines and order types
A pseudoline arrangement in the projective plane P 2 is a set of non-contractible, simple closed curves, any two of which meet in only one point [16]. A pseudoline arrangement is simple if no three pseudolines meet in the same point. Two pseudoline arrangements A and A are isomorphic if there is a self-homeomorphism of P 2 that turns A into A . In what follows, we slightly abuse terminology, and refer to equivalence classes of pseudoline arrangements simply as pseudoline arrangements. A pseudoline arrangement is simply stretchable if and only if it is isomorphic to a simple straight line arrangement in P 2 .
An order type on a set V is a mapping χ : V 3 → {−1, 0, +1} such that the following holds: It is customary to identify an order type χ with its opposite −χ, where all signs are reversed. Order types are one of the several equivalent axiom systems defining oriented matroids of rank three. They are best thought of as a combinatorial abstraction of a point set V in the plane, in which the value of χ(a, b, c) determines whether the three points a, b, c make a left turn (−1), a right turn (+1), or are aligned (0). When χ(a, b, c) = 0 for all triples a, b, c, the order type is said to be in general position, and the corresponding oriented matroid is said to be uniform. An order type χ is realizable if there exists a point set in R 2 with order type χ.
The Folkman-Lawrence topological representation theorem states the equivalence between oriented matroids and classes of topological arrangements [17]. The following special case has been proved in various contexts, in particular by Goodman and Pollack [18] (Remark 1.6, Theorem 1.7 and Corollary 2.11), and later by Knuth [19] (see Section 8, using the terminology of reflection networks, instead of pseudoline arrangements).

Lemma 1.
Order types are in one-to-one correspondence with pseudoline arrangements with a marked face.
Order types in P 2 (which we do not consider in this paper) correspond exactly to pseudoline arrangements. For order types in R 2 , the pseudolines bounding the marked face correspond to the points on the convex hull of the order type. In this bijection, order types in general position are in one-to-one correspondence with simple pseudoline arrangements. Whenever the order type is realizable, the above correspondence is between sets of points and sets of straight lines, and can be obtained by projective duality.
In 1988, Mnëv showed that every semialgebraic set is stably equivalent to the realization space of some oriented matroid of rank three [12]. Furthermore it was shown that the underlying matroid could be made uniform (see also Lemma 4 in Shor [20]). This has the following complexity-theoretic consequence.
The following well-known lemma follows from Theorem 1 and Lemma 1: Order type realizability is ∃R-complete, even when the order type is known to be in general position.
We will use this lemma to prove ∃R-completeness of k-SGE in the next section.

∃R-completeness of k-SGE
We first repeat the reduction from weak rectilinear realizability here and then give a direct proof by reduction from order type realizability.

Reduction from weak rectilinear realizability
An abstract topological graph (AT-graph) is a pair (G, R) where G = (V, E) is a graph and R ⊆ E 2 is a set of pairs of its edges. A straight-line drawing of G is a weak rectinilear realization of (G, R) if every pair of edges that cross in the drawing is contained in R. The complexity of deciding if an AT-graph has a weak rectilinear realization was settled by Kynčl: ). Weak rectilinear realizability is ∃R-complete.
Kynčl proves ∃R-hardness of weak rectilinear realizability [1] by a reduction from the pseudoline stretchability problem. Given an arrangement A of n pseudolines, Kynčl constructs an AT-graph (G, R) with G = (V, E) that admits a weak rectilinear realization if and only if A is stretchable.
This problem is closely related to the k-SGE problem. The following equivalence is analogous to the equivalence given in Theorem 1 of [8]. Given graphs Combining Kynčl's argument with this equivalence yields the following: given an arrangement A of n pseudolines, we can construct a set of k n graphs G i , each on m n vertices, such that the family {G i } i admits a k n -SGE if and only if A is stretchable. Here, k n = Θ(n 4 ) and m n = Θ(n 2 ). By adding an additional k n − m n isolated vertices to each graph G i , we obtain the following: Theorem 3. Given graphs G 1 = (V, E 1 ), . . . , G k = (V, E k ) on n vertices, the decision problem k-SGE is ∃R-complete for k = Ω(n).

Reduction from order type realizability
We will give an alternative proof of the result from the previous section by giving a polynomial time reduction from order type realizability to k-SGE. Fix an order type χ on n elements in general position (i.e., the range of χ is {−1, +1}).
We begin by noting that the convex hull is also well-defined for order types that are not realizable: the (clockwise) convex hull h 1 , . . . , h t of χ is uniquely defined by the property that For an element h i on the convex hull of χ, let S(h i ) be the (hull) surrounding sequence of h i . This is the unique sequence v 1 , . . . , v n−1 of all elements other than  For an element v not on the convex hull of χ, let S(v) be the (internal) surrounding sequence of v. This is the unique (up to cyclic shifts) sequence v 1 , . . . , v n−1 of all elements other than v such that (i) For point sets, this is the cyclic order in which the elements of V \ {v} are encounterd by a counterclockwise ray sweep around v, starting at v 1 . We will usually treat internal surrounding sequences as bi-infinite repetitions . . . , v 1 , . . . , v n−1 , v 1 , . . . . See Fig. 2 for examples of surrounding sequences. (Note that surrounding sequences are different from the local sequences defined by Goodman and Pollack [18]. The latter correspond to sweeps with a line, not a ray.) Proof. We show that we can reconstruct each entry of χ from the surrounding sequences and the convex hull. Consider an entry χ(u, h i , w) of χ. Since h i is on the convex hull, it follows immediately that χ(u, h i , w) = +1 if and only if u occurs before w in S(h i ) (Fig. 3a). Similarly, we have χ(h i , v, w) = +1 if and only if v occurs before w in S(h i ) (Fig. 3b). Since χ(a, b, c) = −χ(c, b, a) for all  triples a, b, c and any χ, we have χ(u, v, h Fig. 3. Cases in the proof of Lemma 3. Next, consider an entry χ(u, v, w) where u, v and w are not on the convex hull of χ. Suppose there is a vertex h i of the convex hull such that u and w both occur before v in S(h i ). Shift S(v) cyclically such that h i is the first element in the sequence. We have χ(u, v, w) = +1 if and only if u occurs before w in S(v) (Fig. 3c). The case where there is vertex h i of the convex hull such that u and w both occur after v in S(h i ) is symmetric.
Finally, suppose that for all vertices h i of the convex hull it holds that v occurs between u and w (or between w and u) in S(h i ). Pick any h i such that u occurs before v and v occurs before w in S(h i ). Shift S(u) cyclically such that h i is the first element in the sequence. We have χ(u, v, w) = −1 if and only if v occurs before w in S(u) (Fig. 3d). This concludes the proof.
The following is a direct consequence from Lemma 1 and point-line duality: Lemma 4. All face markings in a stretchable pseudoline arrangement yield a realizable order type.
Since every arrangement of n pseudolines has at least n triangular faces [21], we can assume by Lemma 4 that the marked face of A is triangular. Equivalently, we may assume that χ has a triangular convex hull. If this is not the case, we can transform it into an order type with triangular convex hull by changing the marked face in polynomial time, while preserving stretchability.
We now have all the ingredients required for the reduction. For each v ∈ V we define the wheel graph W v on V by starting with the cycle S v and connecting v to all vertices in S v . We next create the labeled graph T v by embedding three copies of W v into the interior faces of the frame graph T shown on the left in Fig. 4. We distinguish the vertices of different copies by adding a superscript i to the vertices of copy i. The convex hull h 1 1 , h 1 2 , h 1 3 is embedded onto t 1 , t 4 , t 3 ; the convex hull h 2 1 , h 2 2 , h 2 3 is embedded onto t 2 , t 4 , t 1 ; and the convex hull h 3 Fig. 4. From left to right: the frame grah T , a wheel graph Wv for an order type with convex hull h1, h2, h3, and the graph Tv obtained by mapping Wv onto T .
is embedded onto t 3 , t 4 , t 2 . Fig. 4 shows an example of a wheel graph W v and the resulting graph T v .
Though the T v in the example is maximal planar, this is not always the case. If we extend the example with an additional vertex c such that S(v) = a, h 1 , b, c, h 2 , h 3 , then T v will have a face h 1 1 , h 1 2 , b 1 , c 1 . We do, however, have the following.
Proof. Using symmetry it is easy to verify that both T and W v are 3-connected. We will use Menger's theorem to prove that T v is also 3-connected. Let u be any vertex of W v . From every vertex u in W v there is a path to h 1 , a path to h 2 and a path to h 3 such that the paths share no vertex other than u. This can be seen as follows. If u = v then we can reach each h i in one step. Otherwise, one path traverses the cycle in a clockwise direction, one traverses the cycle in a counterclockwise direction and one goes via v. The same holds for T (which can also be thought of as a wheel graph). It follows immediately that there are three interior pairwise vertex-disjoint paths between every two vertices in T v . Hence, the lemma follows by Menger's theorem.
Since T v is 3-connected, all embeddings of T v are the same up to the choice of the outer face. Let T be the set of all T v . Proof. Suppose that χ is realizable. Let P be a labeled point set that realizes χ and let p(v) be the point in P that corresponds to v in χ. By Lemma 3, the counterclockwise ray sweep around a point p(v) ∈ P encounters the other points of P in the order S(v). Hence, by construction of the wheel graphs, the induced straight-line drawing of each W v on P is plane. A labeled point set whose induced straight-line drawing of each T v is plane can now easily be constructed from three copies of P and affine transformations.
Conversely, suppose that T permits an n-SGE ϕ and consider its convex hull. Note that the convex hull of ϕ corresponds to a mutual face of all T v : if some T v does not have a face that corresponds to the convex hull, then some vertex of T v must have been embedded in the outer face of T v in ϕ, which is impossible by Lemma 5. If t 1 , t 2 , t 3 is the (clockwise) outer face in ϕ, then the point set corresponding to any of the three copies is a realization of χ. This can be seen as follows. If t 1 , t 2 , t 3 is the outer face in this clockwise order, then the triangle h 1 1 , h 1 2 , h 1 3 is also oriented in this clockwise order in ϕ. By Lemma 5, this triangle must form the convex hull of each W 1 v . Hence, any swap of two elements in any surrounding sequence S(v 1 ) in ϕ will induce a crossing in the drawing of W 1 v . It follows that ϕ is consistent with all surrounding sequences and hence χ is realizable. If a face other than t 1 , t 2 , t 3 was chosen to be the outer face in ϕ, say a face bounded by three vertices of copy one, then the point set corresponding to the vertices of copy two (or three: both work) is a realization of χ by a similar argumentation. This concludes the proof. Theorem 3 follows from Lemma 2, Theorem 4, the fact that each T v has size polynomial in n, and the fact that k-SGE belongs to ∃R [5].