Maximum Matchings and Minimum Blocking Sets in \Theta_{6}-Graphs

Maximum Matchings and Minimum Blocking Sets in -Graphs

Therese Biedl David R. Cheriton School of Computer Science,
University of Waterloo, Canada
biedl,abiniaz,virvine,alubiw@uwaterloo.ca
Ahmad Biniaz David R. Cheriton School of Computer Science,
University of Waterloo, Canada
biedl,abiniaz,virvine,alubiw@uwaterloo.ca
Veronika Irvine David R. Cheriton School of Computer Science,
University of Waterloo, Canada
biedl,abiniaz,virvine,alubiw@uwaterloo.ca
Kshitij Jain Borealis AI, Waterloo, Canada
kshitij.jain.1@uwaterloo.ca
Philipp Kindermann Lehrstuhl für Informatik I, Universität Würzburg, Germany
philipp.kindermann@uni-wuerzburg.de
Anna Lubiw David R. Cheriton School of Computer Science,
University of Waterloo, Canada
biedl,abiniaz,virvine,alubiw@uwaterloo.ca
Abstract

-Graphs are important geometric graphs that have many applications especially in wireless sensor networks. They are equivalent to Delaunay graphs where empty equilateral triangles take the place of empty circles. We investigate lower bounds on the size of maximum matchings in these graphs. The best known lower bound is , where is the number of vertices of the graph. Babu et al. (2014) conjectured that any -graph has a perfect matching (as is true for standard Delaunay graphs). Although this conjecture remains open, we improve the lower bound to .

We also relate the size of maximum matchings in -graphs to the minimum size of a blocking set. Every edge of a -graph on point set corresponds to an empty triangle that contains the endpoints of the edge but no other point of . A blocking set has at least one point in each such triangle. We prove that the size of a maximum matching is at least where is the minimum, over all -graphs with vertices, of the minimum size of a blocking set. In the other direction, lower bounds on matchings can be used to prove bounds on , allowing us to show that .

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1 Introduction

One of the many beautiful properties of Delaunay triangulations is that they always contain a perfect matching, as proved by Dillencourt [21]. This is one example of a structural property of a so-called proximity graph. A proximity graph is determined by a set of geometric objects in the plane, such as all discs, or all axis-aligned squares. Given such a set and a finite point set , we construct a proximity graph with vertex set and with an edge if there is an object from that contains and and no other point of . When consists of all discs, then we get the Delaunay triangulation. Proximity graphs are often defined in a more general way, with constraints on how the objects may touch points  and , but this narrow definition suffices for our purposes.

Various structural properties have been proved for different classes of proximity graphs. Another example, besides the perfect matching example above, is that the -Delaunay graph, which is a proximity graph defined in terms of the set of all axis-aligned squares, has the even stronger property of always having a Hamiltonian path [2].

Our paper is about structural properties of -graphs, which are the proximity graphs determined by equilateral triangles with a horizontal edge. More precisely, for any finite point set , define to be the proximity graph of with respect to upward equilateral triangles , define to be the proximity graph of with respect to downward equilateral triangles , and define , the -graph of , to be their union. In particular, has an edge between points and if and only if there is an equilateral triangle with a horizontal side that contains and and no other point of . Such a triangle can be shrunk to an empty triangle that has one of or at a corner, the other point on its boundary, and no points of in its interior.

The graphs and are triangular-distance (or “TD”) Delaunay graphs, first introduced by Chew [18] in 1989. The class of -graphs was first introduced (via a different definition) by Clarkson [19] in 1987 and Keil [24] in 1988, and the equivalence with the above definition was proved by Bonichon et al. [14] in 2010. See Section 1.1 for more information.

We explore two conjectures about -graphs.

Conjecture 1 (Babu et al. [8]).

Every -graph has a perfect matching.

(a)
(b)
(c)
Figure 1: A -graph on points with a perfect matching and a blocking set of size . (a) A perfect matching. The empty triangles corresponding to edges incident to are highlighted. (b) A blocking set of size . Edges have the same color as their blocking point. (c) . All edges are incident to a point of .

See Figure 1 for an example. The best known bound is that every -graph on points has a matching of size at least  minus a small constant—in fact, this bound holds for any planar graph with minimum degree 3 [27], hence for any triangulation and in particular for each of and (modulo the small additive constant)—see Babu et al. [8] for the exact bound of . Our main result is an improvement of this lower bound:

Theorem 1.

For any point set of size , has a matching of size at least .

We prove Theorem 1 in Section 2 using the same technique that has been used for matchings in planar proximity graphs, namely the Tutte-Berge theorem, which relates the size of a maximum matching in a graph to the number of components of odd cardinality after removing some vertices. In our case, this approach is more complicated because -graphs are not planar.

Our second main result relates the size of matchings to the size of blocking or stabbing sets of proximity graphs, which were introduced by Aronov et al. [5] for purposes unrelated to matchings. For a proximity graph defined in terms of a set of objects , we say that a set of points blocks if has a point inside any object from that contains exactly two points of , i.e., the set destroys all the edges of , or equivalently, has no edges between vertices in . Refer to Figure 1. See Section 1.1 for previous results on blocking sets.

For a set of points , let be the minimum size of a blocking set of . Let be the minimum, over all point sets of size , of . It is known that since that is a lower bound for blocking all -graphs of points [12].

Let be the minimum, over all point sets of size , of the size of a maximum matching in . Conjecture 1 can hence be restated as . We relate the parameters and as follows:

Theorem 2.

For any point set of points in the plane, has a matching of size , i.e., .

On the other hand, if for some constants , then .

The two statements in the theorem are proved in Section 3. The idea of using bounds on blocking sets to obtain bounds on matchings is new, and is proved via the Tutte-Berge theorem. Theorem 2 has two consequences. The first is that Theorem 1 implies that . The second consequence is that Conjecture 1 is equivalent to the following:

Conjecture 2.

.

In the remainder of the paper, we explore an approach to obtaining lower bounds on . For to be a blocking set, it must have a point in every empty triangle of . Let  be the maximum number of pairwise internally-disjoint empty triangles of  and let be the minimum, over all point sets of size , of . Clearly, and . Conjecture 1 would be proved if we could show that . However, in Section 4 we give an example of a point set  of size  with , which shows that .

We also explore a previously-studied variant where the empty triangles must be completely disjoint, i.e., even their boundaries must be disjoint. If is such a set, then every empty triangle in corresponds to an edge in and these edges share no endpoint because the triangles are disjoint. Then, corresponds to a strong matching in . Strong matchings were introduced by Ábrego et al. [1, 2] who showed that although Delaunay and -Delaunay graphs have perfect matchings, they need not have strong perfect matchings (for discs and squares, respectively). See the following subsection for further background. Biniaz et al. [12] proved that for any point set of size , has a strong matching of at least edges. We prove an upper bound by giving an example where the maximum strong matching in has edges.

In the final Section 5, we prove some additional bounds on the number of edges, maximum vertex degree, and maximum independent set of -graphs.

1.1 Background

-graphs and TD-Delaunay graphs. The -graph on a set of points in the plane, as originally defined by Clarkson [19] and Keil [24], is a geometric graph with vertex set and edges constructed as follows. For every point , place 6 rays emanating from at angles that are multiples of radians from the positive -axis. These rays partition the plane into cones with apex , which we label in counterclockwise order starting from the positive -axis; see Figure 1(a). Add an edge from to the closest point in each cone , where the distance between the apex and a point in is measured by the Euclidean distance from to the projection of on the bisector of as depicted in Figure 1(a). It is straight-forward to show that this definition of -graphs is equivalent to the definition of . The edges of  come from the odd cones, and the edges of  come from the even cones, so the TD-Delaunay graphs and are known as “half-” graphs.

TD-Delaunay graphs are called TD-Delaunay “triangulations”. In fact, they may fall short of being triangulations. As discussed by Drysdale [22] and Chew [18] (see also [7]), they are plane graphs that consist of a “support hull” which need not be convex, and a complete triangulation of the interior (an explicit proof can be found in [8]). This anomaly is often remedied by surrounding the point set with a large bounding triangle. We will use the same approach later on.

(a)
(b)
Figure 2: The construction of (a) the -graph, and (b) the odd half--graph.

The -graphs, and the more general -graphs which are defined in terms of cones, have some properties that are relevant in a number of application areas. In particular, they are sparse has at most edges [26]—and they are spanners—the ratio (known as the spanning ratio) of the length of the shortest path between any two vertices in , , to the Euclidean distance between the vertices is at most a constant [15, 17, 18, 24]. Because of these properties -graphs have applications in many areas including wireless networking [4, 16], motion planning [19], real-time animation [23], and approximating complete Euclidean graphs [18, 25].

Among theta graphs, has some nice properties that make it suitable for communications in wireless sensor networks. In particular, is the smallest integer for which: {enumerate*}[label=()]

has spanning ratio 2 [14, 17, 15];

the so-called -graph which is a subgraph of that limits the in-degree of each node, is a spanner [20]; and

the half--graph admits a deterministic local competitive routing strategy [16].

Convex Distance Delaunay Graphs. For a set of homothets of a convex polygon, the corresponding proximity graphs are the convex distance Delaunay graphs. This concept has been thoroughly studied, see e.g. [22, 7]. Some of the helper lemmas we need for half--graphs come from more general results that hold for all convex distance Delaunay graphs.

Blocking Sets in Proximity Graphs. Blocking or “stabbing” sets were introduced by Aronov et al. [5] as a more flexible way to represent graphs via proximity. The idea was explored further by Aichholzer et al. [3] who showed that points are sufficient and at least points are necessary to block any Delaunay triangulation with vertices. Biniaz et al. [12] showed that at least points are necessary to block any -graph with  vertices. This bound is tight for -graphs and provides a lower bound on . For results on blocking [higher order] Gabriel graphs, see [6, 13].

Strong Matchings for Proximity Graphs. The idea of “strong matchings” in proximity graphs—i.e., pairwise disjoint objects from each with two points of on the boundary and no points in the interior—was introduced by Ábrego et al. [2, 1] for line segments, rectangles, discs, and squares. They note that strong perfect matchings always exist in the first two cases, but show that they do not always exist for discs (Delaunay graphs) and squares (-Delaunay graphs). In fact, they prove upper bounds of and , respectively, on the size of a strong perfect matching. They also give lower bounds of for discs and for squares. The bound for squares was improved to by Biniaz et al. [12] who also proved lower bounds of for and for .

1.2 Preliminaries

We assume that points are in general position and that no line passing through two points of makes an angle of , or with the horizontal.

Notation. For two points and in the plane, we denote by (resp., by ) the smallest upward (resp., downward) equilateral triangle that has and on its boundary. We say that a triangle is empty if it has no points of in its interior. With these definitions, the -graph has an edge between and if and only if is empty or is empty, in which case we say that the edge is introduced by  or by .

Let be a -graph on point set . We use the following notation:

Furthermore, we define to be the minimum of the corresponding parameter over all -graphs on points.

Properties of -graphs. We need the following two properties of -graphs:

Lemma 1 (Babu et al. [8]).

Let be a set of points in the plane, and let and be any two points in . There is a path between and in that lies entirely in . Moreover, the triangles that introduce the edges of this path also lie entirely in . Analogous statements hold for and .

We remark that this lemma holds more generally for any convex-distance Delaunay graph. The second property we need has been proved in the general setting of convex-distance Delaunay graphs. It generalizes the fact that the (standard) Delaunay triangulation contains the minimum spanning tree with respect to Euclidean distances. We state the result for the special case of equilateral triangles. For any two points and in the plane, define the weight function to be the scaling factor, relative to the unit triangle, of the smallest triangle containing and .

Lemma 2 (Aurenhammer and Paulini [7], Theorem 4).

The minimum spanning tree of points  with respect to the weight function is contained in .

Alternatively, the weight of can be defined to be the area of (see [12]) since that gives the same weight-ordering of edges, which is all that matters to Kruskal’s minimum spanning tree algorithm.

A consequence of Lemma 2 (as noted by Aurenhammer and Paulini in their more general setting) is that the minimum spanning tree of points with respect to the weight function  is contained in both  and , because . In particular, this means that the intersection of the graphs and is connected, as was proved using a different method by Babu et al. [8].

The Tutte-Berge Matching Theorem. Let be a graph and let be an arbitrary subset of vertices of . Removing will split  into a number of connected components. Let denote the number of components of , and let denote the number of odd components of , i.e., components with an odd number of vertices. In 1947, Tutte [28] characterized graphs that have a perfect matching as exactly those graphs that have at most odd components, for any subset . In 1957, Berge [10] extended this result to a formula (today known as Tutte-Berge formula) for the size of maximum matchings in graphs. The following is an alternate way of stating this formula in terms of the number of unmatched vertices, i.e., vertices that are not matched by the matching.

Theorem 3 (Tutte-Berge formula; Berge [10]).

The number of unmatched vertices of a maximum matching in is equal to the maximum over subsets of .

To obtain a lower bound on the size of a maximum matching it suffices, by Theorem 3, to find an upper bound on that holds for any . We will use this approach in our proofs of Theorems 1 and 2. In fact, as in Dillencourt’s proof [21] that Delaunay graphs have perfect matchings we will find an upper bound on that holds for any , i.e., we establish a bound on the toughness of the graph [9].

2 Bounding the Size of a Matching

In this section, we prove Theorem 1. Let be a set of points in the plane and let be the -graph on . We will prove that contains a matching of size at least . As implied by Theorem 3, in order to prove a lower bound on the size of maximum matching in , it suffices to prove an upper bound on that holds for any subset of . Since it is hard to argue about odd components, we will in fact prove an upper bound on . Such a bound applies to because .

Our proof will depend on an analysis of the faces of for which we need some preliminary results. Consider a planar graph with a fixed planar embedding. Such a planar embedding divides the plane into connected regions, called faces. For every face  of , we define its degree as the number of triangles in a triangulation of plus 2; see Figure 3 for some examples. Let denote the set of faces of of degree . An easy counting argument (also used by Biedl et al. [11]) shows that if , then , since a face of degree gives rise to faces in a triangulation of , and any triangulation of has faces.

Figure 3: The notion of degree of a face.

We will utilize the following lemma that Dillencourt used in his proof that every Delaunay triangulation contains a perfect matching. Let denote the subgraph of that is induced by a subset of its vertices.

Lemma 3 (Dillencourt [21], Lemma 3.4).

Let be a triangulated planar graph and let be a subset of vertices of . Then, every face of contains at most one component of .

We aim to apply this result to and . As noted in Section 1.1, the interior faces of and are triangles, but their outer faces need not be the convex hull of . For this reason, and also for Lemma 4 below, we add a set of surrounding points as follows. Find the smallest upward and downward equilateral triangle  and  containing all points of . Let be the region (we will need this definition again in Section 3). Observe that all of the empty triangles that introduce edges of lie in , so adding points outside does not remove any of the edges from the graph. We now place points near the corners of and as follows (see Figure 4): at each corner, place a point in the cone opposite to the cone that contains the triangle, and name the points in such a way that every point of has in cone .

Now fix a set for which we want to bound , and define . Pick an arbitrary representative point from every connected component of , and let be the set of these points, so .

Define and consider its subgraph induced by . Note that the outer face of both and is the hexagon formed by ; we add three graph edges (not straight-line segments) to triangulate the outer face, so that is triangulated. Note that none of the points of (and in particular therefore no points of ) are inside the four newly introduced triangular faces.

Figure 4: Augmentation of : the shaded region is , and .

Let be the number of faces of degree in that contain some point of . We define . Since all faces of are now triangles, Lemma 3 applies and every face of contains at most one component, hence at most one point of . Therefore,

(1)

where is defined in a symmetric manner on graph as the number of faces of degree that contain a point of .

Let be the set of faces of degree in and observe that, since no point of appears in the four triangles outside the hexagon of , we have . As a consequence,

(2)

and similarly .

The crucial insight for getting an improved matching bound is that no component can reside inside a face of degree 3 in both and . Formally, we show:

Figure 5: Illustration for the proof of Lemma 4.
Lemma 4.

We have and .

Proof.

Consider any point , hence . Let and be the faces of and that contain , respectively. It suffices to show that one of and has degree at least 4.

By Lemma 2, the minimum-weight spanning tree of belongs to both and . Find a path  in that connects to some point ; by using such a path with the fewest edges, we ensure that no vertex of except belongs to .

Assume first that is in a cone with even index. Let be the points of that are closest to in cones , respectively; since , such points exist. Refer to Figure 5. By Lemma 1, for every , there exists a path between and in  that lies fully in . By our choices of , no vertex of except is in .

So we have four (not necessarily disjoint) paths in that begin at and end at four points of . These points are distinct because they belong to four different cones of . Furthermore, intermediate points of these paths are not in . This implies that the four points belong to the boundary of the same face of , which is . In consequence, has degree at least 4.

Similarly, if is in a cone with odd index, then has degree at least 4, proving the claim. ∎

Now we have tools to prove an upper bound on the unmatched vertices and, more generally, the toughness of a -graph.

Lemma 5.

For any , we have .

Proof.

Recall that we fixed a set of points in with . So , or equivalently . Combining this with the above inequalities, we get

(by (1))
(by Lemma 4)
(by (2))

Therefore, . In consequence of the Tutte-Berge formula, therefore any maximum matching of has at most unmatched vertices, hence at least matched vertices and . This completes the proof of Theorem 1.

Figure 6: A set of seven points and a subset (which contains the six larger points). The graph contains a singleton-component which lies in a face of degree 3 in (green solid edges) and a face of degree 4 in (blue dashed edges).

Remark. If we knew and (where etc.), then a similar analysis would show , which would imply Conjecture 1 except for a small constant term. However, Figure 6 shows an example where a point lies in a face of degree 3 in and a face of degree 4 in , so our proof-approach cannot be used to prove such a claim.

3 The Relationship Between Blocking Sets and Matchings

In this section, we prove Theorem 2—that a lower bound on the blocking size function implies a lower bound on the size of a maximum matching, and vice versa.

Theorem 2 (a).

For any point set of points in the plane, has a matching of size .

To prove it, we will need the following lemma:

Lemma 6.

For any , we have .

Proof.

Consider a set with points such that . Let be a downward equilateral triangle that strictly encloses all points of . Let be the rightmost point of . Then, lies in cone of . Let be a point strictly inside cone of ; see also Figure 4. Every upward or downward equilateral triangle between and any point of contains the point . Set , and observe that we can block  by using a minimum blocking set  of and adding to it. Since , we have , and cannot be larger than that. ∎

Since , this lemma also shows that , or in other words, that the ‘’ in Conjecture 2 is tight. We are now ready to prove Theorem 2 (a).

Proof.

Consider the -graph on a set of points in the plane. We again use the Tutte-Berge formula (Theorem 3) to prove that contains a matching of size at least . Fix an arbitrary set and consider the connected components of . As in the proof of Theorem 1, fix one representative point in each component, and let be the set of these points.

Now, consider the -graph of only the points in , and let be an edge in it, say it is introduced by . By Lemma 1, there is a path between and in  that is fully contained in . Since and belong to different components of , at least one point of belongs to .

Thus, for any edge in , the triangle that supports that edge contains a point in . Put differently,  blocks , and thus . Furthermore, by Lemma 6 since . Combining this with Theorem 3, it follows that the size of maximum matching in is at least

In particular, if , then , so by integrality . In other words, Conjecture 2 implies Conjecture 1.

Now, we turn to the other half of Theorem 2.

Theorem 2 (b).

Assume that we know that for some constants . Then, .

Before proving this, we would like to point out that a similar result (for and Delaunay graphs) was proved by Aichholzer et al. [3, Theorem 6], and our proof is a modification of theirs. (In fact, the same proof applies to any class of proximity graphs.)

Proof.

Let be a set of points such that , and let be a minimum blocking set of of size . Let be a matching of size at least in . Since is an independent set in , it can contain at most one endpoint of each edge in , as well as all unmatched points, so

Solving for gives . ∎

In particular, if Conjecture 1 holds, then . Hence, and , therefore and Conjecture 2 holds. So Conjecture 1 implies Conjecture 2. As a second consequence, we know that is a valid lower bound on  by Theorem 1, therefore (with ) we have .

4 Other Bounds on , , and .

In this section, we give upper bounds on and . Specifically, we give an example of points for which the maximum number of pairwise internally-disjoint empty triangles is ; this shows that . Then, we give an example on points for which the maximum strong matching has edges; this shows that .

We defined to be the minimum size of a blocking set of any -graph on points because this was relevant for matchings, but it is also interesting to know the maximum number of points that may be needed to block any -graph on points, i.e., to establish bounds on —the maximum, over all points sets of size , of . An easy upper bound on follows from Biniaz et al. [12] who showed that can always be blocked by points placed just above every input point except for the topmost one. By symmetry, can always be blocked by points, and thus, can be blocked by at most points, i.e., . Our final example of this section is a set of points such that , and thus ; this shows that .

(a)
(b)
Figure 7: A point set for which the maximum number of disjoint triangles is . (a) The basic cluster , its -graph, and a set of 3 possible internally-disjoint empty triangles. (b) The final point set formed by repeating . Only some of the empty triangles between clusters are shown.

An upper bound on

Figure 7 shows how to construct a point set of size such that . The point set consists of repeated copies of a cluster of four points arranged as shown in Figure 6(a). Observe that there are 8 empty triangles formed by pairs of points in : 2 for each of the three dashed edges, and 1 for each of the two long blue edges—we call these the “blue triangles”.

Claim 1.

In , there are at most 3 interior-disjoint empty triangles.

Proof.

If neither blue triangle is used, then there are at most 3 interior-disjoint triangles, one for each dashed edge. Using both blue triangles rules out all other empty triangles. Using exactly one blue triangle rules out both empty triangles corresponding to the black dotted edge. ∎

The final configuration consists of  copies of , called clusters, where lies in cone  of all the points of . If we do not use empty triangles determined by pairs of points from different clusters, then by Claim 1 we can get at most 3 empty triangles for each 4 points in for a total of interior disjoint empty triangles. It remains to analyze what happens when we use empty triangles between different clusters.

Consider an empty triangle determined by two points and in different clusters. Then, the points lie in consecutive clusters, say and . Furthermore, one of the points, say , lies at a corner of . We assign the triangle to the cluster of the other point . Observe (see Figure 6(b)) that must be the unique extreme point of its cluster, but point is not unique. The proof that the point set allows at most interior-disjoint empty triangles follows from the following claim.

Claim 2.

For any set of interior-disjoint empty triangles and any , there are at most 3 triangles assigned to or contained in .

Proof.

Consider and suppose that our set contains one between-cluster empty triangle assigned to . By symmetry, we may suppose that this triangle has a corner at a point in ; see, for example, the large yellow triangle in Figure 6(b). This triangle intersects 4 of the empty triangles of , and it is easy to check that there are at most 2 internally-disjoint triangles left.

Next, suppose that we use more than one between-cluster empty triangle assigned to . Then, there must be exactly two such triangles, one with a corner in and one with a corner in . But then, all the empty triangles inside are ruled out. ∎

An upper bound on .

Figure 8 shows how to construct a point set  of size such that . The point set consists of repeated copies of a cluster of five points arranged as shown in Figure 7(a). It is crucial that the two triangles shown in the figure intersect. The final configuration consists of copies of , again called clusters, where lies in cone of all the points of . See Figure 7(c).

(a)
(b)
(c)
Figure 8: A point set for which the maximum strong matching has at most edges. (a) The basic cluster  of 5 points. The dashed lines are guide lines at and . (b) The -graph of , some of the empty triangles, and a strong matching (dashed red). (c) The final point set formed by repeating . Only some of the empty triangles between clusters are shown.

If we do not use empty triangles between clusters, then each cluster has at most two disjoint empty triangles, i.e., at most two strong matching edges, so the matching has at most edges. As in the previous construction, an empty triangle determined by two points and in different clusters must go between consecutive clusters, and one point, say , must lie at a corner of . As before, we assign such a triangle to the cluster containing the other point . The proof that the point set allows at most strong matching edges follows from the following claim.

Claim 3.

For any set of disjoint empty triangles and any , there are at most 2 triangles assigned to or contained in .

Proof.

Consider and suppose that our set contains one between-cluster empty triangle assigned to . By symmetry, we may suppose that this triangle has a corner at a point in ; see, for example, the large yellow triangle in Figure 7(c). There are only 4 points and 5 empty triangles in that are disjoint from the big triangle (these are shown with solid thin lines in the central cluster in Figure 7(c)), and we claim that no two of those are disjoint. In more detail, and referring to the figure, a strong perfect matching would have to match the bottommost point of the cluster with the central point, but the corresponding triangle intersects all the other 4 empty triangles.

Next, suppose that the set contains more than one between-cluster empty triangle assigned to . Then, there must be exactly two such triangles, one with a corner in , and one with a corner in . But then, all the empty triangles inside are ruled out. ∎

A lower bound on .

Figure 9 shows how to construct a set of points with at least pairwise internally-disjoint empty triangles. Start with the triangle which has two points on its boundary, then attach to it copies of the gadget stacked one on top of the other; this gadget adds four points and five interior-disjoint triangles.

Figure 9: A set of points with at least pairwise internally-disjoint empty triangles.

5 Additional Properties of -Graphs

In this section, we prove some addition structural properties of -graphs. First, we are interested in the density, i.e., the number of edges. Clearly, the -graph is connected, hence has at least edges, and this is achieved for example by points on a vertical line. Morin and Verdonschot [26] showed that the expected number of edges (of the -graph of a set of points, chosen randomly, uniformly and independently in a unit square) is . As for the maximum number, an easy argument shows that there are at most edges: For any set of points, the graphs and are planar and contain at most edges each. The edges of a minimum spanning tree belong to both graphs, so their union contains at most edges. We can improve this slightly:

Lemma 7.

Any -graph on points has at most edges.

Proof.

Consider the graphs and . If one of them has an outer face that is not a triangle, then it has at most edges, and re-doing the above analysis gives the bound. If both and have a triangle as outer face, then the vertices on them are necessarily the same three vertices, and the three edges between them belong to both and and form a cycle. Since the minimum spanning tree also belongs to both  and , there are at least edges common to both graphs and again re-doing the analysis gives the bound. ∎

It is worth noting that and actually cannot both have a triangular outer face for since this would contradict Lemma 4: With , the outer face we would have , while since there is only one component of .

Note that the bound is tight for if the three points form a triangle. We do not know whether it is tight for larger . Babu et al. [8] found a set of points whose -graph has edges. We can improve on this and show that the factor ‘5’ in the upper bound is tight.

Lemma 8.

For any , there exists a set of points whose -graph has edges.

Proof.

See Figure 9(a). Start with a set  of points on a vertical line; these have edges between them (black bold). Add 6 surrounding points as in Figure 4. Each of is adjacent to all points of , adding edges. We are free to move (within their respective regions) and can arrange them such that they form an octahedron, adding 12 edges among them (blue dashed). Finally, we have one edge each from and to the topmost/bottommost point of . Hence, in total we have edges. ∎

(a)
(b)
Figure 10: (a) A set of points whose -graph has edges. (b) A set of points whose -graph has minimum degree 7. Not all edges are shown.

Since the number of edges of every -graph with vertices is at most , its total vertex-degree is at most , so some vertex has degree at most 9. In particular, therefore -graphs are 9-degenerate, which implies that they are 10-vertex-colorable (even 10-list-colorable) and have an independent set of size at least .

It remains open whether there are -graphs with minimum degree 9 or even minimum degree 8, but we can construct one with minimum degree 7; see Figure 9(b). We construct our graph by starting with (black bold edges), realized in such a way that each vertex  has cones that are empty (contain no other point). Then, we add 6 surrounding points, arranged so that they form an octahedron (blue dashed edges). With this, each point of obtains another edge in each of its empty cones and hence has degree 7. This gives a graph where all but two vertices have degree 7 or more. Taking two copies of this graph and placing them such that the copies of and become adjacent then gives a -graph of minimum degree 7. Note that more edges appear between the copies, but the minimum degree remains 7.

6 Conclusions and Open Problems

We have improved the lower bound on the size of a matching in any -graph on points to . A main open problem is to prove the conjecture that any -graph has a perfect matching.

We have shown that this conjecture is equivalent to proving that every -graph on points requires at least points to block all its edges. More generally, we proved a relationship between the minimum size of maximum matchings and the minimum size of blocking sets so that any improvement in the lower bound for one of these parameters will also improve the other.

Acknowledgements. This work was done by a University of Waterloo problem solving group. We thank the other participants, Alexi Turcotte and Anurag Murty Naredla, for helpful discussions.

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