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Graphs and Combinatorics

, Volume 32, Issue 3, pp 923–942 | Cite as

Counting Carambolas

  • Adrian Dumitrescu
  • Maarten Löffler
  • André Schulz
  • Csaba D. Tóth
Original Paper

Abstract

We give upper and lower bounds on the maximum and minimum number of geometric configurations of various kinds present (as subgraphs) in a triangulation of n points in the plane. Configurations of interest include convex polygons, star-shaped polygons and monotone paths. We also consider related problems for directed planar straight-line graphs.

Keywords

Convex polygon Star-shaped polygon Monotone path  Plane graph Triangulation Counting 

1 Introduction

We consider plane straight-line graphs (also referred to as planar geometric graphs), where the vertices are points in the plane and the edges are line segments between the corresponding points, no two of which intersect except at common endpoints. According to a classical result of Ajtai et al. [2], the number of plane straight-line graphs on n points in the plane is \(O(c^n)\), where c is a large absolute constant.

Problems in extremal graph theory typically ask for the minimum or maximum number of certain subgraphs, e.g., perfect matchings, spanning trees or spanning paths, contained in a graph of a given order. Here we consider extremal problems in plane straight-line graphs, where the classes of subgraphs are defined geometrically (i.e., membership in the class depends on the coordinates of the vertices). For instance, van Kreveld et al. [16] studied the number of convex polygons (cycles) in geometric triangulations on n points (vertices). They constructed n-vertex triangulations containing \(\Omega (1.5028^n)\) convex polygons, and proved that every triangulation on n points in the plane contains \(O(1.6181^n)\) convex polygons. Dumitrescu and Tóth [9] subsequently sharpened the upper bound to \(O(1.5029^n)\), thereby almost closing the gap between the upper and lower bounds.
Fig. 1

A “carambola” in a triangulation

In this paper we continue this research direction and investigate the multiplicities of other geometrically defined subgraphs present in a geometric triangulation, such as star-shaped polygons and monotone paths. A star-shaped polygon (a.k.a. carambola, see Fig. 1) is a simple polygon P such that there is a (center) point o in its interior with the property that every ray emanating from o intersects the boundary of P in exactly one point. As we will see, star-shaped polygons are closely related to monotone paths.

Let \(\mathbf {u}\in \mathbb {R}^2\setminus \{\mathbf {0}\}\) be a nonzero vector (here \(\mathbf {0}=(0,0)\)). A polygonal path \(\xi =(v_1,v_2,\ldots , v_t)\) is monotone in direction \(\mathbf {u}\), if every line orthogonal to \(\mathbf {u}\) intersects \(\xi \) in at most one point. Equivalently, a path \(\xi \) is monotone in direction \(\mathbf {u}\), if every directed edge of \(\xi \) has a positive scalar product with \(\mathbf {u}\), that is, \(\langle \overrightarrow{v_iv_{i+1}},\mathbf {u}\rangle >0\) for \(i=1,\ldots , t-1\). A special case is an x-monotone path, which is monotone in the horizontal direction \(\mathbf {u} =(1,0)\). A path \(\xi =(v_1,v_2,\ldots , v_t)\) is monotone if it is monotone in some direction \(\mathbf {u}\in \mathbb {R}^2\setminus \{\mathbf {0}\}\). Monotone paths are traditionally used in optimization; a classic example is the simplex algorithm for linear programming, which traces a monotone path on the 1-skeleton of a d-dimensional polytope of feasible solutions.

Counting star-shaped polygons and monotone paths in a triangulation T can be reduced to counting directed cycles and paths, respectively, in some orientation of T. Indeed, let o be a center point and orient every edge ab in T as (ab) iff \(\Delta {oab}\) is a clockwise triangle. Then the number of star-shaped polygons centered at o equals the number of directed cycles. Similarly, let \(\mathbf {u}\in \mathbb {R}^2\setminus \{\mathbf {0}\}\) be a nonzero vector, and orient every edge ab in T as (ab) iff \(\langle \overrightarrow{ab},\mathbf {u}\rangle >0\). Then the number of \(\mathbf {u}\)-monotone paths in T equals the number of directed paths. To see the role played by the geometric constraints in the definition of these special orientations, we also derive bounds on the maximum and minimum number of directed paths in an oriented n-vertex triangulation.

Our results. In Sects. 2 and 3, we derive lower and upper bounds, respectively, on the maximum number of subgraphs of a certain kind that a triangulation on n points can contain. Table 1 summarizes known and new results.
Table 1

Bounds for the maximum number of configurations in an n-vertex plane graph

Configurations

Lower bound

Upper bound

Convex polygons

\(\Omega (1.5028^n)\) [16]

\(O(1.5029^n)\) [9]

Star-shaped polygons

\(\Omega (1.7003^n)\)

\(O(n^3\alpha ^n)\)

Monotone paths

\(\Omega (1.7003^n)\)

\(O(n\alpha ^n)\)

Directed simple paths

\(\Omega (\alpha ^n)\)

\(O(n^2 3^n)\)

Results in row 1 are included for comparison; the bounds in rows 2-4 are proved in the paper. Row 4 concerns directed graphs. \(\alpha =1.8392\ldots \) is the unique real root of the cubic equation \(x^3-x^2-x-1=0\)

In Sect. 4, we study the minimum number of configurations that a triangulation on n vertices can contain. Our asymptotic bounds are summarized in Table 2; more precise estimates are available in the respective section.
Table 2

Bounds for the minimum number of configurations in a triangulation with n vertices

Configurations

Lower bound

Upper bound

Convex polygons

\(\Omega (n)\)

O(n)

Star-shaped polygons

\(\Omega (n)\)

\(O(n^2)\)

Monotone paths

\(\Omega (n^2)\)

\(O(n^{3.17})\)

Directed paths

\(\Omega (n)\)

O(n)

Row 4 concerns directed triangulations

Related work. Previous research studied the maximum number of cycles and spanning trees in triangulations with n vertices. Since cycles and spanning trees, in general, have no geometric attributes, upper bounds on these numbers hold for all edge-maximal planar graph, i.e., combinatorial triangulations. Buchin et al. [3] showed that every triangulation with n vertices contains \(O(2.8928^n)\) simple cycles, and there are triangulations that contain \(\Omega (2.4262^n)\) simple cycles and \(\Omega (2.0845^n)\) Hamiltonian cycles. Buchin and Schulz [4] proved that every n-vertex triangulation contains \(O(5.2852^n)\) spanning trees. These techniques are instrumental for bounding the total number of noncrossing Hamiltonian cycles and spanning trees that n points in the plane admit [10, 15]. Some recent lower bounds on these numbers appear in [7]; see also [1, 8, 10] and the references therein for other related results.

2 Lower Bounds on the Maximum Number of Subgraphs

2.1 Monotone Paths

We construct plane straight-line graphs on n vertices that contain \(\Omega (1.7003^n)\) x-monotone paths; see Fig. 2. Thus by orienting all edges from left to right, we obtain a directed plane straight-line graph that contains \(\Omega (1.7003^n)\) directed paths. The same construction yields lower bounds for a few related subgraphs. By arranging three copies of this graph around the origin in a cyclic fashion as shown in Fig. 5, we obtain a plane straight-line graph that contains \(\Omega (1.7003^n)\) star-shaped polygons. By connecting the leftmost and rightmost vertices by two extra edges to a common vertex, we obtain a plane straight-line graph that contains \(\Omega (1.7003^n)\) monotone polygons, (defined as usual; see e.g., [5, p. 49]).
Fig. 2

Left a graph on \(n=2^\ell +2\) vertices (here \(\ell =3\)), that contains \(\Omega (1.7003^n)\) monotone paths, for n sufficiently large. Right a straight-line embedding of the same graph where the points lie alternately on two circular arcs, while preserving x-monotonicity

Let \(n=2^\ell +2\) for an integer \(\ell \in \mathbb {N}\). We define a plane graph G on n vertices \(V=\{v_1,\ldots , v_n\}\): it consists of a path \(\pi = (v_1,\ldots , v_n)\) and two balanced binary triangulations of the vertices \(\{v_1,\ldots , v_{n-1}\}\) and \(\{v_2,\ldots , v_n\}\), respectively, one on each side of the path; see Fig. 2 (left). Specifically, G contains an edge \((v_i,v_{i+2^k})\), for \(1\le i \le n-2^k\), if and only if \(i-1\) or \(i-2\) is a multiple of \(2^k\). A straight-line embedding is shown in Fig. 2 (right), where the odd and respectively the even vertices lie on two convex polygonal chains with opposite orientations.

Theorem 1

The graph G described in the preceding paragraph has \(\Omega (1.7003^n)\) x-monotone paths.

Fig. 3

The five possible x-monotone paths in the group of \(G_2\)

Proof

We count the number of x-monotone paths in a sequence of subgraphs of G. Let \(G_0\) be the path \(\pi = (v_1,\ldots , v_n)\); and we recursively define \(G_k\) from \(G_{k-1}\) by adding the edges \((v_i,v_{i+2^k})\) for \(i=j2^k+1\) and \(i=j2^k+2\) for \(j=0,1,\ldots , 2^{\ell -k}-1\). The final graph is \(G_\ell \). Denote by \(p_k(v_i)\) the number of x-monotone paths in \(G_k\) that end at vertex \(v_i\). Since every monotone path can be extended to the rightmost vertex \(v_n\), the number of maximal (with respect to containment) monotone paths in \(G_k\) is \(p_k(v_n)\).

We establish the following recurrence relations for \(p_k(v_i)\). The initial values are \(p_k(v_1)=p_k(v_2)=1\) and \(p_k(v_3)=2\) for all \(k=0,\ldots ,\ell \). For \(k=1\) and \(i\ge 3\), we have \(p_1(v_i)=p_1(v_{i-1})+p_1(v_{i-2})\), therefore \(p_1(v_i)=F_i\), where \(F_i\) is the ith Fibonacci number. It is well known that \(F_i= \Theta (\phi ^n)\), where \(\phi =(1+\sqrt{5})/2=1.6180\ldots \), so \(p_1(v_n)=\Theta (\phi ^n)\).

The recurrence for \(p_k(v_i)\), \(k\ge 2\), is more nuanced, due to the asymmetry between the triangulations on the two sides of the path \(\pi \). We partition the edges of graph \(G_k\) into groups, each induced by \(2^{k-1}+2\) consecutive vertices (with respect to \(\pi \)), such that every two consecutive groups share two vertices. Let \(a_i\) denote the first vertex of group i, and let \(b_i\) be the second vertex of group i. Let the edge \((a_i,b_i)\) belong to group i but not to group \(i-1\). We count the number of ways one can route an x-monotone path through a group. A path through group i starts at either \(a_i\) or \(b_i\), and ends at either \(a_{i+1}\) or \(b_{i+1}\). Thus, it is enough to keep track of four different types of paths. We record the number of paths from \(a_i\) or \(b_i\) to \(a_{i+1}\) or \(b_{i+1}\) in a \(2\times 2\) matrix \(M_k\), such that
$$\begin{aligned} M_k\cdot (p_k(a_i),p_k(b_i))^T=(p_k(a_{i+1}),p_k(b_{i+1}))^T. \end{aligned}$$
Once the matrix \(M_k\) is known, we can compute the number of paths by \((p(v_{n-1}),p(v_n))^T=M_k^{(n-2)/2^{k-1}} \cdot (1,1)^T\). By the Perron–Frobenius Theorem [11], \(\lim _{q\rightarrow \infty } M_k^q/\lambda ^q = A\), for some matrix A, and for \(\lambda \) being the largest eigenvalue of \(M_k\). Hence, we have \(\lim _{n\rightarrow \infty }p_k(v_n)=\Theta (\lambda ^{n/2^{k-1}})\) maximal x-monotone paths in \(G_k\).
We now show how to compute the matrices \(M_k\) by induction on k. The matrix \(M_2\) can be easily obtained by hand (see Fig. 3). For computing \(M_k\), \(k\ge 3\), consider an arbitrary group i of size \(2^{k-1}+2\) in \(G_k\). This group is composed of two consecutive groups of \(G_{k-1}\) that share two common vertices, say \(a_j\) and \(b_j\), and two additional edges \(a_ia_{i+1}\) and \(b_ib_{i+1}\). We distinguish two types of paths in group i of \(G_k\): (1) Paths that use only the edges in \(G_{k-1}\). Every such path is the concatenation of two paths, from two consecutive groups of \(G_{k-1}\), with a common endpoint \(a_j\) or \(b_j\). The number of these paths from \(a_i\) or \(b_i\) to \(a_{i+1}\) or \(b_{i+1}\) is precisely \(M_{k-1}^2\). (2) The paths that use \(a_ia_{i+1}\) or \(b_ib_{i+1}\). Since edge \(a_ib_i\) is part of group i, but edge \(a_{i+1}b_{i+1}\) is not, the only possible paths are \((a_i,a_{i+1})\), \((a_i,b_i,b_{i+1})\), and \((b_i,b_{i+1})\); see Fig. 4. Therefore, we can compute the matrices \(M_k\) iteratively as follows:
$$\begin{aligned} M_2:=\begin{pmatrix} 2 &{}\quad 1 \\ 1 &{}\quad 1 \end{pmatrix},\quad \text { and }\quad M_k:=M_{k-1}^2 + \begin{pmatrix} 1 &{}\quad 0 \\ 1 &{}\quad 1 \end{pmatrix}. \end{aligned}$$
Table 3 shows the values \(\lambda ^{1/2^{k-1}}\) for \(k=2,\ldots ,6\). Note that when going from \(k=5\) to \(k=6\), there is no change in \(\lambda ^{1/2^{k-1}}\) up to 8 digits after the decimal point. The precise value of \(\lambda \) for \(k=5\) equals \(\lambda =\frac{1}{2} (4885 + 9 \sqrt{294153})\). \(\square \)
Fig. 4

Schematic drawing of the paths counted by \(M_k\)

Table 3

The asymptotic growth of the number of x-monotone paths in the graphs \(G_k\)

k

2

3

4

5

6

\(\lambda ^{1/2^{k-1}}\)

1.61803

1.69605

1.70034

1.70037

1.70037

Already for \(k=4\) there are \(\Omega (1.7003^n)\) monotone paths

Fig. 5

Cyclic embedding of 3 copies of the graph in Fig. 2. The monotone order becomes a cyclic order

2.2 Star-Shaped Polygons

Use a projective transformation of the plane straight-line graph in Fig. 2 in which the point at infinity in direction (0, 1) moves to the center o of the equilateral triangle in Fig. 5. Use three copies of the resulting graph; the monotone order becomes a cyclic order with respect to the center o. The plane straight-line graph in Fig. 5 has \(\Omega (1.7003^n)\) star shaped polygons, obtained by concatenating the images of any three maximal monotone paths, one in each copy.

2.3 Directed Plane Graphs

We construct n-vertex directed plane straight-line graphs that contain \(\Omega (1.8392^n)\) directed paths; Fig. 6 shows the directed graph and Fig. 7 shows a planar embedding. It is worth noting that the directed paths, however, cannot be extended to a cycle because in a planar embedding the start and end vertex are not in the same face. Similarly, most of the directed paths are not monotone in any direction in a planar embedding.
Fig. 6

There are \(\Theta (\alpha ^n)\) directed paths in this graph. A plane embedding of the graph is depicted in Fig. 7. \(\alpha =1.8392\ldots \) is the unique real root of the cubic equation \(x^3-x^2-x-1=0\)

Fig. 7

A plane embedding of the graph in Fig. 6. The edges are directed from outer circles to inner circles

Denoting by T(i) the number of directed paths ending at vertex \(v_i\), we have \(T(1)=T(2)=1\), \(T(3)=2\), and a linear recurrence relation
$$\begin{aligned} T(i) = T(i-1) + T(i-2) + T(i-3), \quad \text {for } i\ge 4. \end{aligned}$$
The recurrence solves to \(T(i)=\Theta (\alpha ^i)\), where \(\alpha =1.8392\ldots \) is the unique real root of the cubic equation \(x^3-x^2-x-1=0\). Therefore the total number of directed paths, starting at any vertex, is \(\Theta (\alpha ^n)\).

3 Upper Bounds on the Maximum Number of Subgraphs

3.1 Monotone Paths

We start with x-monotone paths in a plane straight-line graph. We prove the upper bound for a broader class of graphs, since some of the operations in our argument may not preserve straight-line edges. A plane monotone graph is a graph embedded in the plane such that every edge is an x-monotone Jordan arc.

Let \(n\in \mathbb {N}\), \(n\ge 3\), and let \(G = (V, E)\) be a plane monotone graph with \(|V|=n\) vertices that maximizes the number of x-monotone paths. We may assume that (i) the vertices have distinct x-coordinates (otherwise we can perturb the vertices without decreasing the number of x-monotone paths) and (ii) the vertices lie on the x-axis (by applying a homeomorphism that affects the y-coordinates). We may also assume that G is fully triangulated (i.e., it is an edge-maximal planar graph), since adding x-monotone edges can only increase the number of x-monotone paths [14].

Label the vertices in V as \(v_1, v_2, \ldots , v_n\), sorted by their x-coordinates. Orient each edge \(\{v_i,v_j\} \in E\) from left to right, i.e., from \(v_i\) to \(v_j\) if \(i < j\). Define the length of an edge \(\overline{v_iv_j}\in E\) by \(\mathrm{len}(\overline{v_iv_j})=|i-j|=j-i\).

Consider an edge \(\overline{v_iv_j} \in E\) that is not on the boundary of the outer face. There are two bounded faces incident to \(\overline{v_iv_j}\), and at least two other vertices \(v_k\) and \(v_l\) that are adjacent to both \(v_i\) and \(v_j\). Suppose that \(i< k,l<j\), and without loss of generality, that \(k<l\) (i.e., \(i< k<l<j\)). The flip operation for the edge \(\overline{v_iv_j}\) replaces \(\overline{v_iv_j}\) by the edge \(\overline{v_kv_l}\); note that this operation preserves planarity but may introduce curved edges.

Lemma 1

If \(i < k<l< j\) as described above, then flipping \(\overline{v_iv_j}\) to \(\overline{v_kv_l}\) produces a plane monotone graph \(G'\) with at least as many x-monotone paths as G (see Fig. 8.)

Fig. 8

The flip operation

Proof

The deletion of edge \(\overline{v_iv_j}\) from G creates a quadrilateral face \((v_i,v_k,v_j,v_l)\). Since \(i < k <l < j\), the new edge \(\overline{v_kv_l}\) can be embedded in the interior of this face as an x-monotone Jordan arc.

Every x-monotone path in G that does not contain \(\overline{v_iv_j}\) is present in \(G'\). Define an injective map from the set of x-monotone paths that traverse \(\overline{v_iv_j}\) in G into the set of x-monotone paths that traverse \(\overline{v_kv_l}\) in \(G'\). To every x-monotone path \(\xi \) that traverses \(\overline{v_iv_j}\) in G, map an x-monotone path \(\xi '\) obtained by replacing edge \(\overline{v_iv_j}\) with the path \((v_i,v_k,v_l,v_j)\) in \(G'\). It follows that \(G'\) contains at least as many x-monotone paths as G. \(\square \)

Note that the flip operation described in Lemma 1 decreases the total length of the edges \(\sum _{e\in E} \mathrm{len}(e)\). We may now assume that among all n-vertex plane monotone graphs with the maximum number of x-monotone paths, G has minimal total edge length. Thus Lemma 1 is inapplicable and we have the following.

Lemma 2

For every interior edge \(\overline{v_iv_j}\in E\), with \(i<j\), there is a triangular face \((v_i,v_j,v_k)\) such that either \(k<i<j\) or \(i<j<k\).

We show next that G contains an x-monotone Hamiltonian path.

Lemma 3

All edges \(\overline{v_iv_{i+1}}\) are present in G.

Proof

Suppose, to the contrary, that there are two nonadjacent vertices \(v_i\) and \(v_{i+1}\). Since G is a triangulation, \(v_iv_{i+1}\) is not a boundary segment, and so there exists an edge that crosses the line segment \(v_iv_{i+1}\). Let \(\overline{v_jv_k}\), \(j<k\), be a longest edge with this property. Since the edge \(\overline{v_jv_k}\) is x-monotone, we have \(j < i < i+1 < k\). The edge \(\overline{v_jv_k}\) is not adjacent to the outer face, since it crosses the segment \(v_iv_{i+1}\) between two vertices. Since edge \(\overline{v_jv_k}\) is interior, by Lemma 2, there is a triangular face \((v_j,v_k,v_l)\) such that either \(l<j<k\) or \(j<k<l\). Without loss of generality, assume that \(j<k<l\). Since there is no vertex in the interior of the face \((v_j,v_k,v_l)\), the boundary of the face has to cross the segment \(v_iv_{i+1}\) twice: that is, \(\overline{v_jv_l}\) crosses the segment \(v_iv_{i+1}\). Since \(j<k<l\), we have \(l-j>k-j\), thus \(\overline{v_jv_l}\) is longer than \(\overline{v_jv_k}\), in contradiction to the assumption that \(\overline{v_jv_k}\) is a longest edge crossing \(v_iv_{i+1}\). \(\square \)

For every pair \(i < j\), let \(V_{ij}\) denote the set of consecutive vertices \({v_i, v_{i+1}, \ldots , v_j}\), and let \(G_{ij} = (V_{ij}, E_{ij})\) be the subgraph of G induced by \(V_{ij}\). Since G is planar, we know that \(|E| \le 3|V| - 6\), and furthermore, that \(|E_{ij}| \le 3|V_{ij}| - 6\) for all subgraphs induced by groups of 3 or more consecutive vertices.

In the remainder of the proof we will apply a sequence of shift operations on G (defined subsequently) that may create multiple edges and edge crossings. Hence, we consider G as an abstract multigraph. However, the operations will maintain the invariant that \(|E_{ij}| \le 3|V_{ij}| - 6\) whenever \(|V_{ij}|\ge 3\).

Let \(i < j < k\) be a triple of indices such that \(\overline{v_iv_j}, \overline{v_iv_k}\in E\). The operation shift(ijk) removes the edge \(\overline{v_iv_k}\) from E, and inserts the edge \(\overline{v_jv_k}\) into E (see Fig. 9). Note that the new edge may already have been present, in this case we insert a new copy of this edge (i.e., we increment its multiplicity by one).
Fig. 9

The operation shift(ijk)

Lemma 4

The operation shift(ijk) does not decrease the number of x-monotone paths in G.

Proof

Clearly, any path that used \(\overline{v_iv_k}\) can be replaced by a path that uses \(\overline{v_iv_j}\) and (the new copy of) \(\overline{v_jv_k}\). \(\square \)

Now, we apply the following algorithm to the input graph G. We process the vertices from left to right, and whenever we encounter a vertex \(v_i\) with outdegree 4 or higher, we identify the smallest index j such that \(v_i\) has an edge to \(v_j\) and the largest index k such that \(v_i\) has an edge to \(v_k\); and then apply shift(ijk). We repeat until there are no more vertices with outdegree larger than 3.

Lemma 5

The algorithm terminates and maintains the following two invariants:
  1. (I1)

    All edges \(\overline{v_iv_{i+1}}\) are present in G with multiplicity one.

     
  2. (I2)

    \(|E_{ij}| \le 3|V_{ij}| - 6\) for all subgraphs induced by \(V_{ij}\), \(i<j\).

     

Proof

Initially, invariant (I1) holds by Lemma 3, and (I2) by planarity. Edges between consecutive vertices are neither removed nor added in the course of the algorithm, consequently (I1) is maintained. To show that (I2) is maintained, suppose the contrary, that there is an operation that increases the number of edges of an induced subgraph \(G'\) above the threshold. Let shift(ijk) be the first such operation. Since the only new edge is \(\overline{v_jv_k}\), the subgraph \(G'\) must contain both \(v_j\) and \(v_k\); and it cannot contain \(v_i\) since the only edge removed is \(\overline{v_iv_k}\). Recall that \(v_j\) was the leftmost vertex that \(v_i\) is adjacent to; and by invariant (I1), we know \(j=i+1\). Therefore, \(G'=G_{jk'}\) for some \(k'\ge k\), and we have \(|E_{jk'}| \ge 3|V_{jk'}| - 5\) after the shift. Since \(v_k\) was the rightmost vertex adjacent to \(v_i\) before the shift, all outgoing edges of \(v_i\) went to vertices in \(V_{jk'}\). The outdegree of \(v_i\) was at least 4 before the shift, hence \(G_{ik'}\) had at least \(3(|V_{ik'}|-1)-5+4 = 3|V_{ik'}| - 4 > 3|V_{ik'}| - 6\) edges, which is a contradiction. \(\square \)

Now, after executing the algorithm, we are left with a multigraph where the outdegree of every vertex is at most 3, and no subgraph induced by \(|V_{i,j}|\ge 3\) consecutive vertices has more than \(3|V_{ij}|-6\) edges. This, combined with invariant (I1), implies that the multiplicity of any edge \(\overline{v_iv_{i+2}}\) is at most one. Thus, for every vertex \(v_i\), the (at most) three outgoing edges go to vertices at distance at least 1, 2, and 3, respectively, from \(v_i\). Denoting by T(i) the number of x-monotone paths that start at \(v_{n-i+1}\), we arrive at the recurrence
$$\begin{aligned} T(i) \le T(i-1) + T(i-2) + T(i-3),\quad \text {for } i \ge 4, \end{aligned}$$
with initial values \(T(1)=T(2)=1\) and \(T(3)=2\). The recurrence solves to \(T(n)=O(\alpha ^n)\) where \(\alpha =1.8392\ldots \) is the unique real root of the cubic equation \(x^3-x^2-x-1=0\). Therefore, every plane monotone graph on n vertices admits \(O(\alpha ^n)\) x-monotone paths. In particular, every plane straight-line graph on n vertices admits \(O(\alpha ^n)\) x-monotone paths.

Since the edges of an n-vertex planar straight-line graph have at most \(3n-6=O(n)\) distinct directions, the number of monotone paths (over all directions) is bounded from above by \(O(n\alpha ^n)\). We summarize our results as follows.

Theorem 2

For every \(n\in \mathbb {N}\), every triangulation on n points contains \(O(\alpha ^n)\) x-monotone paths and \(O(n\alpha ^n)\) monotone paths, where \(\alpha =1.8392\ldots \) is the unique real root of \(x^3-x^2 -x -1=0\).

3.2 Star-Shaped Polygons

Given a plane straight-line graph G on n vertices, the lines passing through the O(n) edges of G induce a line arrangement with \(O(n^2)\) faces. Choose a face f of the arrangement, and a vertex v of G. We show that G contains \(O(\alpha ^n)\) star-shaped polygons incident to vertex v and with a star center lying in f. Indeed, pick an arbitrary point \(o\in f\). Each edge of G is oriented either clockwise or counterclockwise with respect to o (with the same orientation for any \(o \in f\)). Order the vertices of G by a rotational sweep around o starting from the ray \(\overrightarrow{ov}\). Let \(G_{f,v}\) be the graph obtained from G by deleting all edges that cross the ray \(\overrightarrow{ov}\). We can repeat our previous argument for monotone paths for \(G_{f,v}\), replacing the x-monotone order by the rotational sweep order about o, and conclude that G admits \(O(\alpha ^n)\) star-shaped polygons incident to vertex v and with the star center in f. Summing over the O(n) choices for v and the \(O(n^2)\) choices for f, we deduce that G admits \(O(n^3 \alpha ^n)\) star-shaped polygons.

3.3 Directed Simple Paths

Let \(G=(V,E)\) be a directed planar graph. Denote by \(\mathrm{deg}^+(v)\) the outdegree of vertex \(v\in V\); let \(V^+=\{v_1,\ldots , v_\ell \}\) be the set of vertices with outdegree at least 1, where \(1\le \ell \le n\). We show that for every \(v \in V^+\), there are \(O(3^n)\) maximal (with respect to containment) directed simple paths in G starting from v. Each maximal directed simple path can be encoded in an \(\ell \)-dimensional vector that contains the outgoing edge of each vertex \(v\in V^+\) in the path (and an arbitrary outgoing edge if \(v\in V^+\) is not part of the path). The number of such vectors is
$$\begin{aligned} \prod _{i=1}^{\ell } \mathrm{deg}^+(v_i) \le \left( \frac{1}{\ell } \sum _{i=1}^{\ell } \mathrm{deg}^+(v_i)\right) ^\ell < \left( \frac{3n}{\ell } \right) ^\ell \le 3^n, \end{aligned}$$
where we have used the geometric-arithmetic mean inequality, and the fact that by Euler’s formula \(\sum _{i=1}^\ell \mathrm{deg}^+(v_i)\le 3n-6<3n\). We then have maximized the function \(x\rightarrow (3n/x)^x\) over the interval \(1\le x\le n\). Since there are O(n) choices for the starting vertex \(v \in V^+\), and a maximal simple path contains O(n) nonmaximal paths starting from the same vertex, the total number of simple paths is \(O(n^2 \, 3^n)\).

4 Bounds on the Minimum Number of Subgraphs

In this section, we explore the minimum number of geometric subgraphs of a certain kind that a triangulation on n points in the plane can have. We start with some easier results concerning convex polygons, star-shaped polygons, and directed paths (in the next three subsections). The most difficult result (concerning monotone paths) is deferred to the last subsection.

4.1 Convex Polygons

Every n-vertex triangulation has at least \(n-2\) triangular faces, hence \(n-2\) is a trivial lower bound for the number of convex polygons. Hurtado et al. [12] proved that every triangulation contains at least \(\lceil n/2\rceil \) pairs of triangles whose union is convex, and this bound is the best possible. Consequently, every triangulation contains at least \(3n/2 -O(1)\) convex polygons, each bounding one or two faces. The n-vertex triangulations in Fig. 10 (left) contains \(4n- O(1)\) convex polygons, and so this bound is the best possible apart from constant factors. The triangulation consists of the join of two paths, \(P_2 *P_{n-2}\), where the path \(P_{n-2}\) is realized as a monotone zig-zag path. Every convex polygon is either a triangle or the union of two adjacent triangles that share a flippable edge [12].
Fig. 10

There are \(\Theta (n)\) convex polygons and x-monotone paths in the triangulation on the left; it contains \(\Theta (n^4)\) star-shaped polygons and monotone paths. There are \(\Theta (n^2)\) star-shaped polygons and \(\Theta (n^4)\) monotone paths in the triangulation on the right

4.2 Star-Shaped Polygons

Every convex polygon is star-shaped, and so the \(3n/2 -O(1)\) lower bound on the number of convex polygons in a triangulation on n points (from the previous paragraph) also holds for star-shaped polygons. The following averaging argument yields an improved lower bound.

Consider a triangulation \(T=(V,E)\) on n points, k of which are in the interior of the convex hull of V. Then T has \(n+k-2\) bounded (triangular) faces, \(2n+k-3\) edges, \(n+2k-3\) of which are interior edges. Consequently, the average vertex degree in T is \(4+2(k-3)/n\). For each vertex \(v\in V\), every sequence of consecutive faces incident to v forms a star-shaped polygon. The number of such sequences is \(2{\deg (v)\atopwithdelims ()2}+1\) for interior vertices and \({\deg (v)\atopwithdelims ()2}\) for boundary vertices. However, summation over all \(v\in V\) counts every triangular face three times, and every quadrilateral formed by a pair of adjacent faces twice. Consequently, the number of star-shaped polygons formed by the union of faces with a common vertex is at least
$$\begin{aligned}&\sum _{v\in V}{\deg (v) \atopwithdelims ()2} +k - 2(n+k-2) - (n+2k-3) \nonumber \\&\quad \ge n{{4+2(k-3)/n}\atopwithdelims ()2} -3n-3k+7\nonumber \\&\quad \ge \frac{n}{2}\cdot \left( 4+\frac{2(k-3)}{n}\right) \left( 3+\frac{2(k-3)}{n}\right) -3n-3k+7\nonumber \\&\quad \ge 6n+7(k-3)+\frac{2(k-3)^2}{n} -3n-3k+7\nonumber \\&\quad = 3n+4k-14+\frac{2(k-3)^2}{n} \end{aligned}$$
(1)
where we have used Jensen’s inequality in the first step. Since \(k \ge 0\), inequality (1) yields at least \(3n-O(1)\) star-shaped polygons.

Our best lower bound construction is a fan triangulation, with a vertex of degree \(n-1\), shown in Fig. 10 (right); it admits \({{n-1}\atopwithdelims ()2}\) star-shaped polygons.

4.3 Directed Paths

In a directed triangulation, every edge is a directed path of length 1, and the boundary of every triangular face contains at least one path of length two (that can be uniquely assigned to it). Since every triangulation on n points has at least \(2n-3\) edges and at least \(n-2\) triangular faces, it follows that every directed triangulation has at least \(3n-5\) directed paths of length 1 or 2. This bound is the best possible. Indeed, the directed triangulation shown in Fig. 11 has \(3n -5\) directed paths: the \(n-1\) fan edges are directed upward and the remaining edges on the convex hull are directed clockwise or counterclockwise, in an alternating fashion. There are \(2n-3\) paths of length 1, and \(n-2\) paths of length 2, each lying on the boundary of a triangular face, and there are no paths of length 3 or higher.
Fig. 11

There are \(\Theta (n)\) directed paths in this directed planar triangulation

4.4 Monotone Paths

4.4.1 Lower Bound

We first argue that every triangulation contains \(\Omega (n^2)\) monotone paths, since there is a monotone path connecting any pair of vertices. Indeed, this is a corollary of the following lemma applied to triangulations:

Lemma 6

[6, Lemma 1] Let v be a vertex in a plane graph \(G=(V,E)\) where every bounded face with \(k\ge 3\) vertices is a convex k-gon, and the outer face is the exterior of the convex hull of V. Then G contains a spanning tree rooted at v such that all paths starting at v are monotone.

It is also worth noting that the monotone path connecting a pair of vertices, u and v, is not necessarily monotone in the direction \(\overrightarrow{uv}\). Also, two vertices are not always connected by an x-monotone path: a trivial lower bound for the number x-monotone paths is \(\Omega (n)\), since every nonvertical edge is x-monotone. The triangulation \(P_2 *P_{n-2}\) in Fig. 10 (left) is embedded such that the path \(P_{n-2}\) is x-monotone and lies to the right of \(P_2\). With this embedding, it contains \(\Theta (n^2)\) x-monotone paths: every x-monotone path consists of a sequence of consecutive vertices of \(P_{n-2}\), and 0, 1, or 2 vertices of \(P_2\). However, both triangulations in Fig. 10 admit \(\Theta (n^4)\) monotone paths (over all directions).

Triangulations with a polynomial number of monotone paths are also provided by known constructions in which all monotone paths are “short”. Dumitrescu, Rote, and Tóth [6] constructed triangulations with maximum degree \(O(\log n/\log \log n)\) such that every monotone path has \(O(\log n/\log n\log n)\) edges. Moreover, there exist triangulations with bounded vertex-degree in which every monotone path has \(O(\log n)\) edges. These constructions contain polynomially many, but \(\omega (n^4)\), monotone paths.

4.4.2 Upper Bound

We construct a triangulation T of n points containing \(O(n^{2\log 3} \log ^2 n) = O(n^{3.17})\) monotone paths1. A similar construction was introduced in [6] as a stacked polytope in \(\mathbb {R}^3\), where every monotone path on its 1-skeleton has \(O(\log n)\) edges.

Theorem 3

For every \(n\in \mathbb {N}\), there is an n-vertex triangulation that contains \(O(n^{2\log 3}\log ^2 n)=O(n^{3.17})\) monotone paths.

Construction. For every integer \(\ell \ge 0\), we define a triangulation T on \(n=2^\ell +2\) vertices. Refer to Fig. 12. The outer face is a right triangle \(\Delta {oab}\), where o is the origin and \(\angle boa =\frac{\pi }{2}\). The interior vertices are arranged on \(\ell \) circles, \(C_0,C_1,\ldots , C_{\ell -1}\), centered at the origin, with the radii of the circles rapidly approaching 0. We place \(2^i\) points on \(C_i\), in an equiangular fashion, as described below, and so the number of interior points is \(\sum _{i=0}^{\ell -1} 2^i= 2^\ell -1\).
Fig. 12

A schematic illustration of the triangulation T. The radii of the circles \(C_i\), \(i=0,1,\ldots ,\ell -1\), converge to 0 much faster than indicated in the figure

The rays to the points on \(C_i\) are interspersed with the rays to points on all previous layers. Specifically, the \(2^i\) points on circle \(C_i\) are incident to rays emitted from the origin in directions \(\frac{\pi }{4}+\frac{2j-1}{4\cdot 2^{i}} \pi \) for \(j=1,\ldots , 2^i\). The edges of the triangulation are defined as follows. The origin o is connected to all other vertices. Each vertex on circle i is connected to the two vertices of the previous layers that are closest in angular order. The radii of the circles \(C_i\) are chosen recursively for \(i=0,1,\ldots ,\ell -1\), such that the edges that connect a vertex \(v\in C_i\) to vertices \(v'\) and \(v''\) on larger circles are almost parallel to \(ov'\) and \(ov''\), respectively. Specifically, we require that \(\angle vv'o<\pi /2^{\ell +1}\) and \(\angle vv''o<\pi /2^{\ell +1}\) (so these angles are less than the angle between any two consecutive edges incident to o).

Maximal monotone paths. In the argument, we sometimes focus on monotone paths that are maximal (with respect to containment). This is justified by the following easy lemma.

Lemma 7

Let T be a triangulation of a point set S in the plane.
  1. (i)

    The two endpoints of a maximal monotone path in T are vertices of the convex hull \(\mathrm{conv}(S)\).

     
  2. (ii)

    If T contains m maximal monotone paths, and every such path has at most k vertices, then the total number of monotone paths in T is at most \(m{k\atopwithdelims ()2}\).

     

Proof

(i) Let \(\xi =(v_1,\ldots , v_t)\) be a maximal path in T that is monotone in direction \(\mathbf {u}\in \mathbb {R}^2 \setminus \{\mathbf {0}\}\). Suppose that the endpoint \(v_t\) lies in the interior of \(\mathrm{conv}(S)\). Note that the angle between any two consecutive edges of T incident to an interior point of \(\mathrm{conv}(S)\), to \(v_t\) in particular, is less than \(\pi \), since each angle is an interior angle of a triangle. Consequently, there is an edge \(v_tw\) in T such that \(\langle \overrightarrow{v_tw},\mathbf {u}\rangle >0\). Now the path \((v_1,\ldots , v_t,w)\) is monotone in direction \(\mathbf {u}\), and strictly contains \(\xi \), in contradiction with the maximality of \(\xi \).

(ii) Every subpath of a monotone path is also monotone (in the same direction). A path with \(t \ge 2\) vertices has exactly \({t \atopwithdelims ()2}\) subpaths, each determined by the two endpoints, and so the claim follows. \(\square \)

Analysis. We say that a (directed) edge or a path is upward if it is (0, 1)-monotone, and downward if it is \((0,-1)\)-monotone. (The horizontal edge ab is neither upward nor downward). Clearly, every upward path is monotone in direction (0, 1), but a path that contains both upward and downward edges may be monotone in some other direction \(\mathbf {u}\in \mathbb {R}^2\setminus \{\mathbf {0}\}\). Since o is the point with the minimum y-coordinate in T, every maximal upward path starts from o and ends at a or b. Consequently every upward path has at most one vertex on each circle, thus at most \(\ell +2=O(\log n)\) vertices overall. In the remainder of the analysis, we bound the number of upward paths in T, and then show that every monotone path is composed of a small number (bounded by a constant) of upward and downward subpaths.

Lemma 8

The number of maximal upward paths in T is \(O(n^{\log 3})\). The number of upward paths is \(O(n^{\log 3} \log n) =O(n^{1.585})\).

Proof

Let \(\tau _i\) denote the total number of upward paths that start from a point on the circle \(C_i\) and end at a or b. Observe that \(\tau _0=2\) and \(\tau _1=3+3=6\). Recall that by construction, each vertex on \(C_i\) is connected to the two vertices of the previous layers that are closest in angular order (left and right). It follows that
$$\begin{aligned} \tau _i= 2 \sum _{j=0}^{i-1} \tau _j + 2,\quad \text {for } i \ge 1, \end{aligned}$$
(2)
where the term 2 counts the direct edges to a and b from the leftmost and the rightmost points of \(C_i\), respectively.
We now prove that \(\tau _i= 2 \cdot 3^i\) for \(i=0,1,\ldots ,\ell -1\). We proceed by induction on i. The base case \(i=0\) is satisfied as verified above by the value \(\tau _0=2\). For the induction step, assume that the formula holds up to i. According to (2) we have
$$\begin{aligned} \tau _{i+1}= & {} 2 \sum _{j=0}^{i} \tau _j + 2 = 2 \left( 2 \sum _{j=0}^i 3^j \right) +2 = 2 \cdot 2 \cdot \frac{3^{i+1} -1}{2} + 2\\= & {} 2 \cdot 3^{i+1} -2 + 2 = 2 \cdot 3^{i+1}, \end{aligned}$$
as required. Write \(\tau = \sum _{i=0}^{\ell -1} \tau _i\). Recall that \(n=2^\ell +2\), hence
$$\begin{aligned} \tau = \sum _{i=0}^{\ell -1} \tau _i \le 3^\ell \le 3^{\log {n}} = n^{\log {3}}. \end{aligned}$$
(3)
It follows that the number of upward paths starting from vertices on \(\bigcup _{i=0}^{\ell -1} C_i\) and ending at a or b is \(\tau \), and further, that the number of upward paths from o or \(\bigcup _{i=0}^{\ell -1} C_i\) to a or b is \(2\tau \). Since every such path has at most \(\ell +2\) vertices, the total number of upward paths is at most \((\ell +1)\tau =O(n^{\log 3}\log n)\). \(\square \)

Corollary 1

The number of upward paths in T starting from o is \(O(n^{\log 3}) = O(n^{1.585})\).

Proof

Let \(v_0\) be a vertex on \(C_i\), incident to two upward edges \((v_0,v_1)\) and \((v_0,v_2)\). Then all upward paths from o to \(v_0\) lie in the union of two triangles \(\Delta {o v_0 v_1} \cup \Delta {o v_0 v_2}\). The subgraph of T lying in (the interior or on the boundary of) each of these triangles is isomorphic to the analogue of T on \(2^{\ell -i}+2\) vertices, with the same upward-downward orientations. As shown in the proof of Lemma 8, there are \(O(3^{\ell -i})\) upward paths from o to \(v_0\) in each of the triangles \(\Delta {o v_0 v_1}\) and \(\Delta {o v_0 v_2}\), and so T contains \(O(3^{\ell -i})\) upward paths from o to \(v_0\). Summing over all vertices of T, the total number of upward paths starting from o is
$$\begin{aligned} O\left( \sum _{i=0}^{\ell -1} 2^i \cdot 3^{\ell -i} \right) = O\left( 3^\ell \sum _{i=0}^{\ell -1} \left( \frac{2}{3} \right) ^i \right) = O\left( 3^\ell \right) = O\left( 3^{\log n}\right) = O\left( n^{\log 3}\right) , \end{aligned}$$
as claimed. \(\square \)

Lemma 8 also yields the following.

Corollary 2

The number of monotone paths in T composed of one upward path and one downward path is \(O(n^{2\log 3} \log ^2 n) = O(n^{3.17})\).

It remains to consider monotone paths in T that cannot be decomposed into one upward path and one downward path.2 Every such path changes vertical direction (upward vs downward) at least twice.

Lemma 9

The total number of monotone paths in T that change vertical direction at least twice is \(O(n^{2\log 3} \log ^2 n) = O(n^{3.17})\).

Proof

We first characterize the monotone paths that change vertical direction at least twice, showing that in fact they change vertical direction at most three times; we then derive an upper bound on their number.

Let \(\xi \) be a maximal monotone path in T that changes vertical direction at least twice. Refer to Fig. 13. By Lemma 7(i), the endpoints of \(\xi \) are vertices of the outer face \(\Delta {oab}\). One of the endpoints of \(\xi \) is a or b, where all upward edges point to a or b, respectively. Consequently, \(\xi \) changes vertical directions at a vertex in the interior of \(\Delta {oab}\) from downward to upward. That is, \(\xi \) contains a subpath \(\xi '=(v_1,v_2,v_3)\) such that \((v_1,v_2)\) is downward, \((v_2,v_3)\) is upward, and \(v_2\ne o\). By symmetry, we can assume that \(v_1\) and \(v_3\) are on the right and left sides of the ray \(\overrightarrow{ov_2}\), respectively. By construction, \(v_1v_3\), \(ov_1\), and \(ov_3\) are edges of T, so \(\Delta {o v_1 v_3}\) is a 3-cycle in T.

Since both \(\overrightarrow{v_1v_2}\) and \(\overrightarrow{v_2v_3}\) are \(\mathbf {u}\)-monotone, for some vector \(\mathbf {u}\ne \mathbf {0}\), the orthogonal vector \(\mathbf {u}^\perp \) lies between the directions of \(\overrightarrow{v_2v_1}\) and \(\overrightarrow{v_2v_3}\). By construction, these directions are within \(\pi /2^{\ell +1}\) from the directions of \(\overrightarrow{ov_1}\) and \(\overrightarrow{ov_3}\), respectively. For every such vector \(\mathbf {u}\), vertex a is \(\mathbf {u}\)-minimal and b is \(\mathbf {u}\)-maximal, and so \(\xi \) is a path from a to b. \(\square \)

The edges of \(\xi \) directly preceding and following \(\xi '=(v_1,v_2,v_3)\) have a crucial role in determining the edges in \(\xi \setminus \xi '\). The proof of Lemma 9 relies on the following two claims.
Fig. 13

Two schematic pictures of a subpath \(\xi '=(v_1,v_2,v_3)\) in T such that the edge \((v_1,v_2)\) is downward, the edge \((v_2,v_3)\) is upward, and \(v_2\ne o\). Left A maximal monotone path \(\xi \) containing \(\xi '\), where \(\xi \setminus \xi '\) lies in the exterior of \(\Delta {o v_1 v_3}\). Right A maximal monotone path \(\xi \) containing \(\xi '\), where part of \(\xi \setminus \xi '\) lies inside \(\Delta {o v_1 v_3}\)

Claim 1

If \(\xi \) does not change vertical direction at \(v_1\), then \(v_1\) uniquely determines the part of \(\xi \) preceding \(v_1\). Similarly, if \(\xi \) does not change vertical direction at \(v_3\), then \(v_3\) uniquely determines the part of \(\xi \) following \(v_3\).

Proof

By symmetry, it is enough to prove the second statement. If \(v_3=b\), the proof is complete. Assume that \(v_3\ne b\), and label the portion of \(\xi \) starting at \(v_3\) by \((v_3,v_4,\ldots ,v_t)\). Recall that by construction, from every interior vertex v, there are two upward edges lying on opposite sides of the ray \(\overrightarrow{ov}\).

We show by induction on \(j=3,\ldots ,t\) that the edge \((v_j,v_{j+1})\) is upward, it lies on the left of \(\overrightarrow{ov_j}\), and the other upward edge starting from \(v_j\) is incident to a vertex on or to the right of \(\overrightarrow{ov_1}\). In the base case, \(j=3\), and by assumption \((v_3,v_4)\) is upward. The upward edge on the right side of \(\overrightarrow{ov_3}\) cannot enter the interior of \(\Delta {o v_1 v_3}\) and so it must go to a vertex on or to the right of \(\overrightarrow{ov_1}\). Thus this edge is not \(\mathbf {u}\)-monotone, and so \((v_3,v_4)\) must be the upward edge that leaves \(v_3\) on the left side of \(\overrightarrow{ov_3}\).

Assume now \(3<j<t\) and the induction hypothesis holds for \(v_{j-1}\). By construction, the downward edges staring from \(v_j\) are almost parallel to \(\overrightarrow{v_jo}\), and so they are not \(\mathbf {u}\)-monotone. By induction, an upward edge \((v_{j-1},r)\) is incident to a vertex on or to the right of \(\overrightarrow{ov_1}\). Since the two upward edges \((v_{j-1},v_j)\) and \((v_{j-1},r)\) are incident to the same triangle of T, it follows that \((v_j,r)\) is an edge in T. Hence, the upward edge on the right side of \(\overrightarrow{ov_j}\) also goes to a vertex on or to the right of \(\overrightarrow{ov_1}\), and so it is not \(\mathbf {u}\)-monotone. Consequently, the \(\mathbf {u}\)-monotone path \(\xi \) leaves \(v_j\) on the unique upward edge on the left side of \(\overrightarrow{ov_j}\). \(\square \)

Claim 2

If \(\xi \) arrives at \(v_1\) on an upward edge, then \(v_3=b\) or \(\xi \) leaves \(v_3\) on an upward edge. Similarly, if \(\xi \) leaves \(v_3\) on a downward edge, then \(v_1=a\) or \(\xi \) arrives at \(v_1\) on a downward edge.

Proof

By symmetry, it is enough to prove the second statement. Assume that \(\xi \) leaves \(v_3\) on a downward edge \((v_3,v_4)\). Then the directions of \((v_3,v_4)\) and \((v_3,v_2)\) are both within \(\pi /2^{\ell +1}\) from \(\overrightarrow{v_3o}\). Thus \(\mathbf {u}^\perp \) is within \(\pi /2^{\ell +1}\) from \(\overrightarrow{ov_3}\). This, in turn, implies that none of the upward edges going into \(v_1\) is \(\mathbf {u}\)-monotone, and the claim follows.

\(\square \)

We continue with the proof of Lemma 9. If the path \(\xi \) changes vertical direction at neither \(v_1\) nor \(v_3\), then Claim 1 implies that \(\xi \) is composed of one upward path and one downward path, contradicting our assumption that \(\xi \) has no such decomposition.

Assume that the path \(\xi \) changes vertical direction at \(v_1\) or \(v_3\). Without loss of generality, assume that \(\xi \) changes vertical direction at \(v_3\), i.e., \(\xi \) leaves \(v_3\) on a downward edge. By Claims 1 and 2, the part of \(\xi \) preceding \(v_1\) is uniquely determined. Note also that \(\mathbf {u}^\perp \) is within \(\pi /2^{\ell +1}\) from \(\overrightarrow{ov_3}\).

We distinguish two cases based on whether the edge \((v_3,v_4)\) of \(\xi \) following \(v_3\) lies in the exterior of the triangle \(\Delta {o v_1 v_3}\) or in its interior; then we estimate the number of such maximal paths \(\xi \) and their subpaths.

Case 1: \(v_4\) lies in the exterior of \(\Delta {o v_1 v_3}\); refer to Fig. 13 (left). Then the vertices of \(\xi \) following \(v_4\) are uniquely determined, analogously to Claim 1. We have O(n) choices for \(v_2\) (which determines the triple \((v_1,v_2,v_3)\)), and \(O(\log n)\) choices for \(v_4\). Consequently, T contains \(O(n \log n)\) maximal monotone paths \(\xi \) of this type. The length of any such path is \(O(\log n)\), and by Lemma 7(ii), the total number of monotone paths of this type is \(O(n \log ^3 n)\).

Case 2: \(v_4=o\) or \(v_4\) lies in the interior of \(\Delta {o v_1 v_3}\); refer to Fig. 13 (right). Then the path \(\xi \) reaches the \(\mathbf {u}\)-maximal vertex of \(\Delta {o v_1 v_3}\), namely o. The portion of \(\xi \) inside \(\Delta {o v_1 v_3}\) is a downward path from \(v_3\) to o. The portion of \(\xi \) from o to b must be an upward path. In summary, the maximal monotone path \(\xi \) is composed of four upward or downward paths: A downward path from a to \(v_2\), an upward edge \((v_2,v_3)\), a downward path from \(v_3\) to o, and an upward path from o to b.

Let us count the number of maximal monotone paths \(\xi \) of this type. By Lemma 8, there are \(O(n^{\log 3})\) choices for the portion of \(\xi \) from o to b, and \(O(n^{\log 3})\) independent choices for the portion from \(v_3\) to o. Since the degree of \(v_3\) is \(O(\log n)\), we have \(O(\log n)\) choices for vertex \(v_2\). Finally, the portion from a to \(v_2\) is uniquely determined by \(v_2\) by Claim 1. This gives an \(O(n^{2\log 3}\log n)\) bound on the number of maximal monotone paths of this type. The length of any such path is \(O(\log n)\).

By Lemma 7(ii), the total number of monotone paths of this type is \(O(n^{2\log 3} \log ^3 n)\). By counting the subpaths of the maximal monotone paths \(\xi \) directly, we can reduce this bound by a logarithmic factor. If a subpath of \(\xi \) changes vertical directions twice, then its two endpoints must lie in the first and last portion of \(\xi \), respectively. By Corollary 1, there are \(O(n^{\log 3})\) choices for the portion of \(\xi \) from o to b and for all subpaths of \(\xi \) incident to o. The length of this last portion is \(O(\log n)\), and so the number of its subpaths incident to \(v_2\) is \(O(\log n)\). Consequently, the total number of monotone paths of this type is \(O(n^{2\log 3}\log ^2 n)\).

Summing over both cases, it follows that the number of monotone paths that change vertical directions at least twice is \(O(n^{2\log 3}\log ^2 n)\), as claimed.

Proof of Theorem 3

The triangulation T contains \(O(n^{\log 3}\log n)\) upward paths by Lemma 8, and \(O(n^{2\log 3}\log ^2 n)\) paths composed of one upward and one downward piece by Corollary 2. Any other monotone path \(\xi =(v_1,\ldots ,v_t)\) changes vertical direction (upward vs downward) at least twice. By Lemma 9, T contains \(O(n^{2\log 3}\log ^2 n)\) such paths. Consequently, the total number of monotone paths in T is \(O(n^{2\log 3}\log ^2 n) = O(n^{3.17})\), as claimed. \(\square \)

5 Conclusion

We have derived estimates on the maximum and minimum number of star-shaped polygons, monotone paths, and directed paths that a (possibly directed) triangulation of n points in the plane can have. Our results are summarized in Tables 1 and 2 of Sect. 1. Closing or narrowing the gaps between the upper and lower bounds remain as interesting open problems. The gaps in the last row of Table 1 and 2nd row of Table 2 are particularly intriguing.

Footnotes

  1. 1.

    Throughout this paper, all logarithms are in base 2.

  2. 2.

    Such paths were inadvertently overlooked in the analysis from [13].

Notes

Acknowledgments

A. Dumitrescu was supported in part by NSF grant DMS-1001667. M. Löffler was supported by the Netherlands Organization for Scientific Research (NWO) under grant 639.021.123. Research by Tóth was supported in part by NSERC (RGPIN 35586) and NSF (CCF-1423615). This work was initiated at the workshop “Counting and Enumerating Plane Graphs,” which took place at Schloss Dagstuhl in March, 2013.

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Copyright information

© Springer Japan 2015

Authors and Affiliations

  • Adrian Dumitrescu
    • 1
  • Maarten Löffler
    • 2
  • André Schulz
    • 3
  • Csaba D. Tóth
    • 4
    • 5
  1. 1.Department of Computer ScienceUniversity of Wisconsin-MilwaukeeMilwaukeeUSA
  2. 2.Department of Computing and Information SciencesUtrecht UniversityUtrechtThe Netherlands
  3. 3.LG Theoretische InformatikFernUniversität HagenHagenGermany
  4. 4.Department of MathematicsCalifornia State University NorthridgeLos AngelesUSA
  5. 5.Department of Computer ScienceTufts UniversityMedfordUSA

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