Counting Plane Graphs: Flippability and its Applications^{†}^{†}thanks: Work on this paper by Micha Sharir and Adam Sheffer was partially supported by Grant 338/09 from the Israel Science Fund. Work by Micha Sharir was also supported by NSF Grant CCF0830272, by Grant 2006/194 from the U.S.Israel Binational Science Foundation, and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University. Work by Csaba D. Tóth was supported in part by NSERC grant RGPIN 35586. Research by this author was conducted at ETH Zürich. Emo Welzl acknowledges support from the EuroCores/EuroGiga/ComPoSe SNF grant 20GG21_134318/1. Part of the work on this paper was done at the Centre Interfacultaire Bernoulli (CIB), EPFL, Lausanne, during the Special Semester on Discrete and Computational Geometry, Fall 2010, and was supported by the Swiss National Science Foundation.
Abstract
We generalize the notions of flippable and simultaneously flippable edges in a triangulation of a set of points in the plane to socalled pseudosimultaneously flippable edges. Such edges are related to the notion of convex decompositions spanned by .
We prove a worstcase tight lower bound for the number of pseudosimultaneously flippable edges in a triangulation in terms of the number of vertices. We use this bound for deriving new upper bounds for the maximal number of crossingfree straightedge graphs that can be embedded on any fixed set of points in the plane. We obtain new upper bounds for the number of spanning trees and forests as well. Specifically, let denote the maximum number of triangulations on a set of points in the plane. Then we show (using the known bound ) that any element point set admits at most crossingfree straightedge graphs, spanning trees, and forests. We also obtain upper bounds for the number of crossingfree straightedge graphs that have , fewer than , or more than edges, for any constant parameter , in terms of and .
1 Introduction
A crossingfree straightedge graph is an embedding of a planar graph in the plane such that the vertices are mapped to a set of points in the plane and the edges are pairwise noncrossing line segments between pairs of points in . (Segments are allowed to share endpoints.) By Fáry’s classical result [10], such an embedding is always possible. In this paper, we fix a labeled set of points in the plane, and we only consider planar graphs that admit a straightedge embedding with vertex set . By labeled we mean that each vertex of the graph has to be mapped to a unique designated point of . Analysis of the number of plane embeddings of planar graphs in which the set of vertices is not restricted to a specific embedding, or when the vertices are not labeled, can be found, for example, in [16, 21, 32].
A triangulation of a set of points in the plane is a maximal crossingfree straightedge graph on (that is, no additional straight edges can be inserted without crossing some of the existing edges). Triangulations are an important geometric construct which is used in many algorithmic applications, and are also an interesting object of study in discrete and combinatorial geometry (recent comprehensive surveys can be found in [7, 17]).
Improving the bound on the maximum number of triangulations that any set of points in the plane can have has been a major research theme during the past 30 years. The initial upper bound of [2] has been steadily improved in several paper (e.g., see [8, 25, 29]), culminating with the current record of due to Sharir and Sheffer [26]. Other papers have studied lower bounds on the maximal number of triangulations (e.g., [1, 9]), and upper or lower bounds on the number of other kinds of planar graphs (e.g., [5, 6, 23, 24]).
Every triangulation of contains the edges of the convex hull of , and the remaining edges of the triangulation decompose the interior of the convex hull into triangular faces. Assume that contains points, of which are on the convex hull boundary and the remaining points are interior to the hull (we use this notation throughout). By Euler’s formula, every triangulation of has edges ( hull edges, common to all triangulations, and interior edges, each adjacent to two triangles), and bounded triangular faces.
Edge Flips.
Edge flips are simple operations that replace one or several edges of a triangulation with new edges and produce a new triangulation. As we will see in Section 3, edge flips are instrumental for counting various classes of subgraphs in triangulations. In the next few paragraphs, we review previous results on edge flips, and propose a new type of edge flip. We say that an interior edge in a triangulation of is flippable, if its two adjacent triangles form a convex quadrilateral. A flippable edge can be flipped, that is, removed from the graph of the triangulation and replaced by the other diagonal of the corresponding quadrilateral, thereby obtaining a new triangulation of . An edge flip operation is depicted in Figure 1(a), where the edge is flipped to the edge . Already in 1936, Wagner [34] has shown that any unlabeled abstract triangulation (in this case, two triangulations are considered identical if we can relabel and change the planar embedding of the vertices of the first triangulation, to obtain the second triangulation) can be transformed into any other triangulation (with the same number of vertices) through a series of edgeflips (here one uses a more abstract notion of an edge flip). When we deal with a pair of triangulations over a specific common (labeled) set of points in the plane, there always exists such a sequence of flips, and this bound is tight in the worst case (e.g., see [3, 19]). Moreover, there are algorithms that perform such sequences of flips to obtain some “optimal” triangulation (typically, the Delaunay triangulation; see [12] for example), which, as a byproduct, provide an edgeflip sequence between any specified pair of triangulations of .
How many flippable edges can a single triangulation have? Given a triangulation , we denote by the number of flippable edges in . Hurtado, Noy, and Urrutia [19] proved the following lower bound.
Lemma 1.1
[19] For any triangulation over a set of points in the plane,
Moreover, there are triangulations (of specific point sets of arbitrarily large size) for which this bound is tight.
To obtain a triangulation with exactly flippable edges (for an even ), start with a convex polygon with vertices, triangulate it in some arbitrary manner, insert a new point into each of the resulting bounded triangles, and connect each new point to the three hull vertices that form the triangle containing . Such a construction is depicted in Figure 2. The resulting graph is a triangulation with vertices and exactly flippable edges, namely the chords of the initial triangulation.
Next, we say that two flippable edges and of a triangulation are simultaneously flippable if no triangle of is incident to both edges; equivalently, the quadrilaterals corresponding to and are interiordisjoint. See Figure 1(b) for an illustration. Notice that flipping an edge cannot affect the flippability of any edge simultaneously flippable with . Given a triangulation , let denote the size of the largest subset of edges of , such that every pair of edges in the subset is simultaneously flippable. The following lemma, improving upon an earlier weaker bound in [13], is taken from Souvaine, Tóth, and Winslow [30].
Lemma 1.2
[30] For any triangulation over a set of points in the plane, .
Galtier et al. [13] show that this bound is tight in the worst case, by presenting a specific triangulation in which at most edges are simultaneously flippable.
Pseudosimultaneously flippable edge sets.
A set of simultaneously flippable edges in a triangulation can be considered as the set of diagonals of a collection of interiordisjoint convex quadrilaterals. We consider a more liberal definition of simultaneously flippable edges, by taking, within a fixed triangulation , the diagonals of a set of interiordisjoint convex polygons, each with at least four edges (so that the boundary edges of these polygons belong to ). Consider such a collection of convex polygons , where has edges, for . We can then retriangulate each independently, to obtain many different triangulations. Specifically, each can be triangulated in ways, where is the th Catalan number (see, e.g., [31, Section 5.3]). Hence, we can get different triangulations in this way. In particular, if a graph (namely, all the edges of are edges of ) does not contain any diagonal of any (it may contain boundary edges though) then is a subgraph of (at least) distinct triangulations. An example is depicted in Figure 1(c), where by “flipping” (or rather, redrawing) the diagonals of the highlighted quadrilateral and pentagon, we can get different triangulations (including the one shown), and any subgraph of the triangulation that does not contain any of these diagonals is a subgraph of these ten triangulations. We say that a set of interior edges in a triangulation is pseudosimultaneously flippable (psflippable for short) if after the deletion of these edges every bounded face of the remaining graph is convex, and there are no vertices of degree 0. Notice that all three notions of flippability are defined within a fixed triangulation of (although each of them gives a recipe for producing many other triangulations).
Our results.
In Section 2, we derive a lower bound on the size of the largest set of psflippable edges in a triangulation, and show that this bound is tight in the worst case. Specifically, we have the following result.
Lemma 1.3 (psflippability lemma)
Let be a set of points in the plane, and let be a triangulation of . Then contains a set of at least psflippable edges. This bound is tight in the worst case.
Table 1 summarizes the bounds for the minimum numbers of the various types of flippable edges in a triangulation.
We also relate psflippable edges to convex decompositions of . These are crossingfree straightedge graphs on such that (i) they include all the hull edges, (ii) each of their bounded faces is a convex polygon, and (iii) no point of is isolated. See Figure 3 for an illustration.
Graph type  Lower bound  Previous  New upper  In the form 

upper bound  bound  
Plane Graphs  [1]  [22, 26]  
Spanning Trees  [9]  [6, 26]  
Forests  [9]  ) [6, 26] 
Counting plane graphs: New upper bounds.
In Section 3, we use Lemma 1.3 to derive several upper bounds on the numbers of planar graphs of various kinds embedded as crossingfree straightedge graphs on a fixed labeled set . For a set of points in the plane, we denote by the set of all triangulations of , and put . Similarly, we denote by the set of all crossingfree straightedge graphs on , and put . We also let and denote, respectively, the maximum values of and of , over all sets of points in the plane.
Since a triangulation of has fewer than edges, the trivial upper bound holds for any point set . Recently, Razen, Snoeyink, and Welzl [22] slightly improved the upper bound on the ratio from down to . We give a more significant improvement on the ratio with an upper bound of . Combining this bound with the recent bound [26], we get . We provide similar improved ratios and absolute bounds for the numbers of crossingfree straightedge spanning trees and forests (i.e., cyclefree graphs). Table 2 summarizes these results^{2}^{2}2Uptodate bounds for these and for other families of graphs can be found in http://www.cs.tau.ac.il/~sheffera/counting/PlaneGraphs.html (version of November 2010)..
We also derive similar ratios for the number of crossingfree straightedge graphs with exactly edges, with at least edges, and with at most edges, for . For the case of crossingfree straightedge graphs with exactly we obtain the bound^{3}^{3}3In the notations , , and , we neglect polynomial factors.
where . Figure 4 contains a plot of the base of the exponential factor multiplying in this bound, as a function of .
Notation.
Here are some additional notations that we use.

For a triangulation and an integer , let denote the number of interior vertices of degree in .

Given two crossingfree straightedge graphs and over the same point set , we write to indicate that every edge in is also an edge in .

Similarly to the case of edges, the hull vertices (resp., interior vertices) of a set of points in the plane are those that are part of the boundary of the convex hull of (resp., not part of the convex hull boundary).

We only consider point sets in general position, that is, no three points in are collinear. For upper bounds on the number of graphs, this involves no loss of generality, because the number of graphs can only increase if collinear points are slightly perturbed into general position.
Separable edges. Let be an interior vertex in a convex decomposition of . Following the notation in [27], we call an edge incident to in separable at if it can be separated from the other edges incident to by a line through (see Figure 5, where the separating lines are not drawn). Equivalently, edge is separable at if the two angles between and its clockwise and counterclockwise neighboring edges (around ) sum up to more than . Following [19], we observe the following easy properties, both of which materialize in Figure 5.

If is an interior vertex of degree in , its three incident edges are separable at , for otherwise would have been a reflex vertex of some face.

An interior vertex of degree or higher can have at most two incident edges which are separable at (and if it has two such edges they must be consecutive in the circular order around ).
2 The size of psflippable edge sets
In this section, we establish the psflippability lemma (Lemma 1.3 from the introduction). We restate the lemma for the convenience of the reader.
Lemma 1.3
(psflippability lemma) Let be a set of points in the plane, and let be a triangulation of . Then contains a set of at least psflippable edges. This bound is tight in the worst case.
Proof . Starting with the proof of the lower bound, we apply the following iterative process to . As long as there exists an interior edge whose removal does not create a nonconvex face, we pick such an edge and remove it. When we stop, we have a crossingfree straightedge graph , all of whose bounded faces are convex; that is, we have a locally minimal convex decomposition of . Note that all original hull edges are still in , and that every interior vertex of has degree at least (recall the general position assumption).
Note that every edge of is separable at one or both of its endpoints, for we can remove any other edge and the graph will continue to have only convex faces (see Figure 6). We denote by the number of edges of , and by the number of its interior edges. Recalling properties (i) and (ii) of separable edges, we have and
(1) 
where is the number of interior vertices of degree in , and is the number of interior vertices of degree at least in with exactly edges separable at . Notice that
The estimate in (1) may be pessimistic, because it doubly counts edges that are separable at both endpoints (such as the one in Figure 6(c)). To address this possible overestimation, denote by the number of edges that are separable at both endpoints, to which we refer as doubly separable edges, and rewrite (1) as
(2) 
Denoting by the number of bounded faces of , we have, by Euler’s formula,
(the expression in the parentheses on the left is the number of faces in , and the expression in the parentheses on the right is the number of edges), or
(3) 
Let , for , denote the number of interior faces of degree in . By doubly counting the number of edges in , and then applying (3), we get
or
(4) 
The number of edges that were removed from is , because a face of of degree must have had diagonals that were edges of . This number is therefore
We next derive a lower bound for the righthand side of (5). For this, we transform into another graph as follows. We first subdivide each doubly separable edge of at its midpoint, say, and add the subdivision point as a new vertex of (e.g., see the vertex in Figure 7). We now modify as follows. We take each vertex of degree in and surround it by a triangle, by connecting all pairs of its three neighbors. Notice that some of these neighbors may be new subdivision vertices, and that some of the edges of the surrounding triangle may already belong to . For example, see Figure 7, where the edges are added around the vertex and the edges are added around the vertex . Next we take each interior vertex with two separable edges at and complete these two edges into a triangle by connecting their other endpoints, each of which is either an original point or a new subdivision point; here too the completing edge may already belong to . For example, see the edge in Figure 7, induced by the two separable edges of the vertex . We then take the resulting graph and remove each vertex of degree and its three incident edges; see the reduced version of in Figure 7. A crucial and easily verified property of this transformation is that the newly embedded edges do not cross each other, nor do they cross old edges of .
The number of bounded faces of the new graph is at least , which is the number of triangles that we have created, and the number of its interior vertices is . Also, still has hull edges. Using Euler’s formula, as in (3) and (4) above, we have . Combining the above, we get
or
Hence, the righthand side of (5) is at least
In other words, the number of edges that we have removed from is at least . On the other hand, we always have and . Substituting these trivial bounds in (5) we get at least psflippable edges. This completes the proof of the lower bound.
It is easily noticed that only flippable edges of could have been removed in the initial pruning stage. Hurtado, Noy, and Urrutia [19] present two distinct triangulations that contain exactly flippable edges (one of those is depicted in Figure 2). These triangulations cannot have a set of more than psflippable edges. Therefore, there are triangulations for which our bound is tight in the worst case. Similarly, for point sets in convex position, all interior edges form a set of psflippable edges, showing that the other term in the lower bound is also tight in the worst case.
Remark. The proof of Lemma 1.3 actually yields the slightly better bound
That is, for the bound to be tight, every interior vertex of degree or higher must have two incident edges separable at (note that this condition holds vacuously for the triangulation in Figure 2).
Convex decompositions.
The preceding analysis is also related to the notion of convex decompositions, as defined in the introduction. Urrutia [33] asked what is the minimum number of faces that can always be achieved in a convex decomposition of any set of points in the plane. Hosono [18] proved that every planar set of points admits a convex decomposition with at most (bounded) faces. For every , GarcíaLopez and Nicolás [15] constructed element point sets that do not admit a convex decomposition with fewer than faces. By Euler’s formula, if a connected crossingfree straightedge graph has vertices and edges, then it has faces (including the exterior face). It follows that for convex decompositions, minimizing the number of faces is equivalent to minimizing the number of edges. (For convex decompositions contained in a given triangulation, this is also equivalent to maximizing the number of removed edges, which form a set of psflippable edges.)
Lemma 1.3 directly implies the following corollary. (The bound that it gives is weaker than the bound in [18], but it holds for every triangulation.)
Corollary 2.1
Let be a set of points in the plane, so that its convex hull has vertices, and let be a triangulation of . Then contains a convex decomposition of with at most of arbitrarily large size, and triangulations for which these bounds are tight. edges. Moreover, there exist point sets convex faces and at most
3 Applications of psflippable edges to graph counting
In this section we apply the psflippability lemma (Lemma 1.3) to obtain several improved bounds on the number of crossingfree straightedge graphs of various kinds on a fixed set of points in the plane.
3.1 The ratio between the number of crossingfree straightedge graphs and the number of triangulations
We begin by recalling some observations already made in the introduction. Let be a set of points in the plane. Every crossingfree straightedge graph in is contained in at least one triangulation in . Additionally, since a triangulation has fewer than edges, every triangulation contains fewer than crossingfree straightedge graphs. This immediately implies
However, this inequality seems rather weak since it potentially counts some crossingfree straightedge graphs many times. More formally, given a graph contained in distinct triangulations of , we say that has a support of , and write . Thus, every graph will be counted times in the preceding inequality.
Recently, Razen, Snoeyink, and Welzl [22] managed to break the barrier by overcoming the above inefficiency. However, they obtained only a slight improvement, with the bound . We now present a more significant improvement, using a much simpler technique that relies on the psflippability lemma.
Theorem 3.1
For every set of points in the plane, of which are on the convex hull,
Proof . The exact value of is easily seen to be
(6) 
because every graph appears times in the sum, and thus contributes a total of to the count. We obtain an upper bound on this sum as follows. Consider a graph and a triangulation , such that . By Lemma 1.3, there is a set of psflippable edges in .^{4}^{4}4Here we implicitly assume that is even. The case where is odd is handled in the exact same manner, since a constant change in the size of does not affect the asymptotic bounds. Let denote the set of edges that are in but not in , and put . Removing the edges of from yields a convex decomposition of which still contains and whose nontriangular interior faces have a total of missing diagonals. Suppose that there are such faces, with diagonals respectively, where . Then these faces can be triangulated in ways, and each of the resulting triangulations contains . We always have , for any , as is easily verified, and so . (Equality occurs when all the nontriangular faces of are quadrilaterals.)
Next, we estimate the number of subgraphs for which the set is of size . Denote by the set of edges of that are not in , and assume that the convex hull of has vertices. Since there are edges in any triangulation of , . To obtain a graph for which , we choose any subset of edges from , and any edges from (the edges of that will not belong to ). Therefore, the number of such subgraphs is at most .
We can thus rewrite (6) to obtain
If , we get , we have . To complete the proof, we note that when . . If
For a lower bound on , we consider the double chain configurations, presented in [14] (and depicted in Figure 8). It is shown in [14] that, when is a double chain configuration, and (actually, only the lower bound on is given in [14]; the upper bound appears in [1]). Thus, we have (for this set , so has no real effect on the asymptotic bound of Theorem 3.1).
For another lower bound, consider the case where is in convex position. In this case we have , and (see [11]). Hence, , whereas the upper bound provided by Theorem 3.1 is in this case. Informally, the (rather small) discrepancy between the exact bound in [11] and our bound in the convex case comes from the fact that when is large, the faces of the resulting convex decomposition are likely to have many edges, which makes substantially larger than . It is an interesting open problem to exploit this observation to improve our upper bound when is large.
3.2 The number of spanning trees and forests
Spanning Trees.
For a set of points in the plane, we denote by the set of all crossingfree straightedge spanning trees of , and put . Moreover, we let .
Buchin and Schulz [6] have recently shown that every crossingfree straightedge graph contains spanning trees, improving upon the earlier bound of due to Ribó Mor and Rote [23, 24]. We thus get for every set of points in the plane. The bound from [6] cannot be improved much further, since there are triangulations with at least spanning trees [23, 24]. However, the ratio between and can be improved beyond that bound, by exploiting and overcoming the same inefficiency as in the case of all crossingfree straightedge graphs; that is, the fact that some spanning trees may get multiply counted in many triangulations.
We now derive such an improved ratio by using psflippable edges. The proof goes along the same lines of the proof of Theorem 3.1.
Theorem 3.2
For every set of points in the plane,
Proof . The exact value of is
Consider a spanning tree and a triangulation , such that . As in Theorem 3.1, let be a set of psflippable edges in . (Here we do not exploit the alternative bound of on the size of .) Also, let denote the set of edges that are in but not in , and put . Thus, as argued earlier, .
Next, we estimate the number of spanning trees for which the set is of size . First, there are ways to choose the edges of that does not use. We next contract the edges of that were chosen to be in (which will result in having some parallel edges, and possibly also loops) and then remove the remaining edges of . This produces a nonsimple graph with vertices and fewer than edges (recall that by Euler’s formula, contains at most edges). Let denote the set of vertices of , and let denote the degree in of a point . As shown in [6, 23], the number of spanning trees in a graph (not necessarily planar or simple) is at most the product of the vertex degrees in . Thus, the number of ways to complete the tree is at most
(where we have used the inequality of means for the first inequality). Hence, there are fewer than spanning trees with . However, when is large, it is better to use the bound from [6] instead.^{5}^{5}5This is not quite correct: When is close to the former bound is smaller (e.g., it is for ), but we do not know how to exploit this observation to improve the bound.
We thus get, for a threshold parameter that we will set in a moment,
The terms in the first sum over increase when , so the sum is at most times its last term. Using Stirling’s formula, we get that for , the last term in the first sum is . Since this also bounds the second sum, we get
as asserted. (The optimal parameter was computed numerically.)
Combining the bound just obtained with [26] implies
Corollary 3.3
.
Remark. It would be interesting to refine the bound in Theorem 3.2 so that it also depends on , as in Theorem 3.1. An extreme situation is when is in convex position (in which case ). In this case it is known that and (see [11]), so the exact ratio is only . This might suggest that when is large the ratio should be considerably smaller, but we have not pursued this in this paper.
Forests.
For a set of points in the plane, we denote by the set of all crossingfree straightedge forests (i.e., cyclefree graphs) of , and put . Moreover, we let . Buchin and Schulz [6] have recently shown that every crossingfree straightedge graph contains forests (improving a simple upper bound of observed in [1]). Following the approach of [6], we combine the bounds for spanning trees (just established) and for plane graphs with a bounded number of edges (established in Section 3.3 below), to obtain the following result.
Theorem 3.4
For every set of points in the plane,
Proof . We define a forest to be a forest that has connected components. The number of forests of a set is denoted by . Since any spanning tree has edges, every forest has edges. One way to bound is by counting the number of plane graphs with edges. This number is bounded in Theorem 3.6 (from the following subsection), where the parameter in that theorem is equal to ; let us denote this bound as . On the other hand, every forest is obtained by deleting edges from a spanning tree. This allows us to bound the number of forests in terms of . Using Theorem 3.2 we get the bound ; denote this bound as . To bound , we evaluate . A numerical calculation shows that the maximum value is obtained for , and the theorem follows since .
As in the previous cases, we can combine this with the bound [26] to obtain
Corollary 3.5
.
Consider once again the case where consists of points in convex position. In this case we have and (see [11]), so the exact ratio is , again suggesting that the ratio should be smaller when is large.
3.3 The number of crossingfree straightedge graphs with a bounded number of edges
In this subsection we derive upper bounds for the number of crossingfree straightedge graphs on a set of points in the plane, with some constraints on the number of edges. Specifically, we bound the number of crossingfree straightedge graphs with exactly edges, with at most edges, and with at least edges. The first variant has already been used in the preceding subsection for bounding the number of forests.
Crossingfree straightedge graphs with exactly edges.
We denote by the set of all crossingfree straightedge graphs of with exactly edges, and put . The following theorem, whose proof goes along the same lines of the proof of Theorem 3.1, gives a bound for
Theorem 3.6
For every set of points in the plane and ,
where
and
(7) 
See Figure 4 for a plot of the base as a function of .
Proof . The exact value of is
where , the support of , is defined as in the case of general crossingfree straightedge graphs treated in Section 3.1. We obtain an upper bound on this sum as follows. Consider a graph and a triangulation , such that . By Lemma 1.3, there is a set of psflippable edges in . Let denote the set of edges that are in but not in , and put . As in the preceding proofs, we have .
Next, we estimate the number of subgraphs for which the set is of size . Denote by the set of edges of that are not in . As argued above, . To obtain a graph for which , we choose any edges from (the edges of that will not belong to ), and any subset of edges from . If , there are no such graphs and we ignore these values of . The number of ways to pick the edges from is at most This implies that
(8)  
(As already noted, when , only the terms for which are taken into account.)
As in the preceding subsection, it suffices to consider only the largest term of the sum. For this, we consider the quotient of the th and st terms (ignoring the notation, which will not affect the exponential order of growth of the terms), which is
To simplify matters, we put and . Moreover, since we are only looking for an asymptotic bound, and are willing to incur small multiplicative errors within the notation, we may ignore the two terms in the numerator when is sufficiently large; we omit the routine algebraic justification of this statement. The above quotient then becomes (approximately) , which is larger than 1 whenever
with given in (7). A simple calculation shows that and for . In other words (and rather unsurprisingly), the index attaining the maximum does indeed lie in the range where the two binomial coefficients in the corresponding terms in (8) are both well defined (nonzero).
Now that we have the largest term of the sum in (8), we obtain
Using Stirling’s approximation, we have
as asserted.
Crossingfree straightedge graphs with at most edges.
For a set of points in the plane and a constant , we denote by the set of all crossingfree straightedge graphs of with at most edges, and put . The bound for in Theorem 3.6 helps us to determine the bound for .
Theorem 3.7
Proof . We begin by noticing that