Drawable and Forbidden Minimum Weight Triangulations (extended abstract) William Lenharty
Giuseppe Liottaz
Abstract A graph is minimum weight drawable if it admits a straight-line drawing that is a minimum weight triangulation of the set of points representing the vertices of the graph. In this paper we consider the problem of characterizing those graphs that are minimum weight drawable. Our contribution is twofold: We show that there exist in nitely many triangulations that are not minimum weight drawable. Furthermore, we present non-trivial classes of triangulations that are minimum weight drawable, along with corresponding linear time (real RAM) algorithms that take as input any graph from one of these classes and produce as output such a drawing. One consequence of our work is the construction of triangulations that are minimum weight drawable but none of which is Delaunay drawable|that is, drawable as a Delaunay triangulation.
1 Introduction and Overview Recently much attention has been devoted to the study of combinatorial properties of well-known geometric structures|often referred to as geometric graphs in the computational geometry literature(see, e.g., [32, 31])|such as Delaunay triangulations, minimum spanning trees, Gabriel graphs, relative neighborhood graphs, -skeleton graphs, and rectangle of in uence graphs. This interest has been motivated in part by the importance of these structures in numerous application areas, including computer graphics, pattern recognition, computational morphology, communication networks, numerical analysis, computational biology, and GIS. Geometric graphs are 2-dimensional straight-line drawings that satisfy some additional geometric constraints. Thus the problem of analyzing the combinatorial properties of a given type of geometric graph naturally raises the question of the characterization of those graphs which admit the given type of straightline drawing. This, in turn, leads to the investigation of the design of ecient algorithms for computing such a drawing when one exists. Although these questions are far from being resolved in general, many partial answers have appeared in the literature. We give two such examples below. A Gabriel graph (also called Gabriel drawing in the graph drawing literature) is a straight-line drawing such that two vertices u and v are adjacent if and only if the disk having u and v as antipodal points does not contain any other vertex of the drawing. Trees that admit a Gabriel drawing are characterized in [4]. Lubiw and Sleumer [27] show that every maximal outerplanar graph is Gabriel drawable. In [21] Research supported in part by by the EC ESPRIT Long Term Research Project ALCOM-IT under contract 20244. Department of Computer Science, Williams College, Williamstown, MA 01267.
[email protected] Dipartimento di Informatica e Sistemistica, Universita di Roma `La Sapienza', via Salaria 113, I-00198 Roma, Italia.
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the characterization is extended to all outerplanar graphs. The area required by Gabriel drawings is investigated in [26]. Delaunay triangulations are planar straight-line drawings with all internal faces triangles and such that three vertices form a face if and only of the the disk passing through them does not contain any other vertex of the triangulation. To our knowledge, no complete combinatorial characterization of Delaunay drawable triangulations has been given to date. Di Battista and Vismara [9] give a characterization based on a non-linear system of equations involving the angles in the triangulation. Dillencourt has shown that all Delaunay drawable triangulations are 1-tough, and have perfect matchings [13], and that all maximal outerplanar graphs are Delaunay drawable [12]. Dillencourt and Smith [14] show that any triangulation without chords or non facial triangles is Delaunay drawable. A survey on the problem of drawing a graph as a given type of geometric graph can be found in [7]. In this paper we consider a special type of straight-line drawings, called minimum weight drawings, which have applications in areas including computational geometry and numerical analysis. Let C be a class of graphs, let P be a set of points in the plane, and let G be a graph such that 1. G has vertex set P , 2. the edges of G are straight-line segments connecting pairs of points of P , 3. G 2 C , and 4. the sum of the lengths of the edges of G is minimized over all graphs satisfying 1{3. We call such a graph G a minimum weight representative of C . Given a graph G 2 C , we say that G has a minimum weight drawing for class C if there exists a set P of points in the plane such that G is a minimum weight representative of C . For example, a minimum spanning tree of a set P of points is a connected, straight-line drawing that has P as vertex set and minimizes the total edge length. So, letting C be the class of all trees, a tree G has a minimum weight drawing if there exists a set P of points in the plane such that G is isomorphic to a minimum spanning tree of P . A minimum weight triangulation of a set P is a triangulation of P having minimum total edge length. Letting C be the class of all planar triangulations, a planar triangulation G has a minimum weight drawing if there exists a set P of points in the plane such that G is isomorphic to a minimum weight triangulation of P . The problem of testing whether a tree admits a minimum weight drawing is essentially solved. Monma and Suri [30] proved that each tree with maximum vertex degree at most ve can be drawn as a minimum spanning tree of some set of vertices by providing a linear time (real RAM) algorithm. In the same paper it is shown that no tree having at least one vertex with degree greater than six can be drawn as a minimum spanning tree. As for trees having maximum degree equal to six, Eades and Whitesides [15] showed that it is NP-hard to decide whether such trees can be drawn as minimum spanning trees. The representability of trees as minimum spanning trees in three-dimensional space was studied in [25]. Surprisingly, little seems to be known about the problem of constructing a minimum weight drawing of a planar triangulation. Moreover, it is still not known whether computing a minimum weight triangulation of a set of points in the plane is an NP-complete problem (see Garey and Johnson [16]). Several papers have been published on this last problem, either providing partial solutions, or giving ecient approximation heuristics. See, for example, the work by Meijer and Rappaport [29], Lingas [24], Keil [17], Dickerson et al. [10], Kirkpatrick [18], Aichholzer et al. [1], Cheng and Xu [5], and Dickerson and Montague [11], Levcopoulos and Krznaric [23, 22]. The problem examined in this paper is that of characterizing those triangulations that admit a minimum weight drawing. In [20, 19] it was shown that all maximal outerplanar triangulation are minimum weight 2
drawable and a linear time (real RAM) drawing algorithm for computing a minimum weight drawing of these graphs was given. In this paper we describe several other non-trivial classes of minimum weight drawable triangulations, and present in nite families of triangulations that are not minimum weight drawable. Furthermore, we show that the class of graphs that can be drawn as Delaunay triangulations and the class of graphs that can be drawn as minimum weight triangulation do not coincide. This is the rst result showing the combinatorial di erence between Delaunay triangulations and minimum weight triangulations. Note that the study of geometric di erences between minimum weight triangulations and Delaunay triangulations in order to compute good approximations of the former has a long tradition (see, e.g., [28, 18, 23]). More precisely, we: 1. Exploit the relationship between locally minimum weight triangulations and minimum weight triangulations to construct classes of triangulations that are not minimum weight drawable. 2. Exhibit an in nite class of triangulations that can be drawn as minimum weight triangulations but not as Delaunay triangulations. 3. Establish the minimum weight drawability of several non-trivial classes of triangulations, including wheels, spined triangulations, and nested triangulations. (For de nitions see Section 2) 4. Investigate the combinatorial structure of minimum weight triangulations by means of the notion of skeleton of a triangulation, that is the graph induced by the set of its internal vertices. We show that any forest can arise as the skeleton of a minimum weight drawable triangulation. 5. Present linear time, real RAM algorithms that accept as input a minimum weight drawable triangulation of one of the types mentioned above and produce as output a minimum weight drawing of that triangulation. Our algorithmic techniques generalize drawing strategies that have been devised in recent years to compute proximity drawings of graphs (see, e.g., [3, 8, 21]). The rest of this paper is organized as follows. Preliminary de nitions and basic properties of minimum weight triangulations are in Section 2. Classes of triangulations that do not have a minimum weight drawing are presented in Section 3. In the same section we compare minimum weight drawable and Delaunay forbidden triangulations. Section 4 is devoted to presenting classes of minimum weight drawable triangulations and their corresponding algorithms. Open problems are discussed in Section 5. For reasons of space, some of the proofs have been sketched or omitted in this extended abstract.
2 Preliminaries We rst discuss some terminology and de ne classes of triangulations that are of interest in this paper. We then recall some basic properties of minimum weight triangulations. We assume familiarity with basic computational geometry, graph drawing and graph theory concepts. For further details see [2, 32, 6]. A triangulation of a nite set P of points in the plane consists of a maximal sized set of non-intersecting segments connecting pairs of points in P . A graph G is called a triangulation if it is isomorphic to a triangulation of some nite set of points in the plane. Many classes of graphs arise from using some geometric constraint to de ne edges on a set of points in the plane. Three such classes, Gabriel graphs, Delaunay triangulations and minimum weight triangulations, have already been mentioned. Given a certain class C of graphs de ned by such a method, we can de ne 3
a graph G as being C drawable if G is isomorphic to some member of C . Thus in the rest of the paper we will refer to certain triangulations as being, for example, Delaunay drawable, minimum weight drawable, locally minimum weight drawable (see below for a de nition) and so forth. We make particular use of some special types of triangulations. A fan consists of a cycle a; v1 ; v2 ; : : : ; v along with edges from a to each vertex v . The vertex a is called the apex of the fan, and the vertices v are called the neighbors of a. The edges av are called the radial edges of the fan. An example of a fan is shown in Figure 4 (a). A wheel graph (or wheel, for brevity) is a triangulation consisting of a cycle and a single vertex c, called the center of the wheel, adjacent to all vertices on the cycle. The edges from c to the vertices of the cycle are called the spokes of the wheel. The class of k-nested triangulations can be recursively de ned as follows. A three-cycle is a 0-nested triangulation. For k > 0, a k-nested triangulation is de ned to be one having a triangular outer face, the deletion of which results in a k ? 1-nested triangulation. Figure 1 (b) shows a 3-nested triangulation. Another family of triangulation we will study in this paper is the class of k-spined triangulations. Each element of the class is obtained by k-spining K4 . Let a ; a1 ; a2 be the vertices on the outer face and let c be the interior vertex of K4 . k-spining K4 consists of replacing edge ca2 with a path of k vertices such that each new vertex is adjacent to both a0 and a1 . The resulting triangulation is a k-spined triangulation. Figure 1 (c) and (d) show a 1-spined and a 2-spined triangulation, respectively. The Delaunay drawability of spined triangulations is studied in [12]. The skeleton of a triangulation is the graph induced by the set of its internal vertices. For example, the skeleton of a maximal outerplanar graph is the empty graph, the skeleton of a wheel graph consists of just one vertex (the center of the wheel). Figure 1 (a) shows a triangulation whose skeleton is a tree. In the gure, the skeleton is highlighted. n
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Figure 1: (a) A Triangulation whose skeleton is a tree, (b) A 3-nested triangulation, (c) A 1-spined triangulation, and (d) A 2-spined triangulation. We list some basic properties of minimum weight triangulations.
Property 2.1 A minimum weight triangulation of a set S of points in the plane always includes the shortest edge between two points of S and the all edges of the convex hull of S .
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Property 2.2 The minimum weight triangulation of a set S of points in the plane need not contain the
minimum spanning tree of S as a subgraph.
If T is a triangulation, and abd and bcd are two triangles of T which form a convex quadrilateral, then the operation of replacing edge bd with edge ac is called an edge ip. A triangulation T is locally minimum weight if no single edge ip reduces the weight of T .
Property 2.3 If a triangulation cannot be drawn as a locally minimum weight triangulation, then it cannot be drawn as a minimum weight triangulation.
In this paper we will explore the relationship between minimum weight drawable triangulations and Delaunay drawable triangulations. To our knowledge, no complete combinatorial characterization of Delaunay drawable triangulations has been to date given [12, 14, 13]. Di Battista and Vismara [9] give a characterization based on a non linear system of equations involving the angles in the triangulation. Dillencourt [12, 13] gives a set of necessary conditions for Delaunay drawability of a triangulation. In the following theorem, T is a triangulation, P a subset of the set S of the vertices of T , jP j is the cardinality of P , and T ? P is the graph obtained by removing P (and all attached edges) from T .
Theorem 2.1 [12, 13] If a triangulation T can be drawn as a Delaunay triangulation, then for any given P S the following two conditions hold. 1. T is 1-tough, that is , T ? P has at most jP j components. 2. T ? P contains at most jP j ? 2 components that do not contain any vertex of the outer face of T .
3 Forbidden Triangulations In this section we consider the question of whether every triangulation is minimum weight drawable. In Section 3.1 we show classes of minimum weight forbidden triangulations, i.e. triangulations that are not minimum weight drawable. Interestingly, the triangulations in these classes are also Delaunay forbidden, that is, they cannot be drawn as Delaunay triangulations. This observation leads us to comparing minimum weight and Delaunay forbidden triangulations. In Section 3.2 we show an in nite family of triangulations that are minimum weight drawable and Delaunay forbidden.
3.1 Minimum weight forbidden triangulations We construct classes of triangulations that are not locally minimum weight drawable, and therefore, by Property 2.3, not minimum weight drawable. Our rst lemma gives a set of necessary conditions that a triangulation must satisfy in order to be a locally minimum weight triangulation. But rst some notation: To any vertex v and incident face f of a triangulation T , we associate a variable = (v; f ) called the angle of f at v; collectively these variables are called the angles of T . Any straight-line drawing of T determines values for these variables, and thus certain relations among the angles of T must hold. If the triangulation is to be locally minimum weight, additional relations among the magnitudes of the angles must hold. We de ne the magnitude of by m( ) = 1 if is obtuse and m( ) = 0 otherwise.
Lemma 3.1 Let T be a locally minimum weight triangulation. Then there exists an assignment of values to the angles of T such that the following condition holds:
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For each pair of faces f1 and f2 of T that share an edge e, the two angles 1 in f1 and 2 in f2 opposite e satisfy m( 1 ) + m( 2) 1.
Sketch of Proof: It suces to note that in any locally minimum weight triangulation, any two triangles which share an edge must form a quadrilateral for which that edge is not the longer of the two diagonals. Thus the two angles opposite that edge cannot both be obtuse. 2 We can thus use Property 2.3 and Lemma 3.1 to identify forbidden triangulations.
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Figure 2: Two examples of triangulations that cannot be drawn as minimum weight triangulations.
Lemma 3.2 The triangulations of Figure 2 (a) and (b) are minimum weight forbidden. Sketch of Proof: The proof consists of showing that there is no assignment of values to the angles of the
two triangulations that can satisfy Lemma 3.1. Hence, neither triangulation is locally minimum weight drawable, and so, by Property 2.3, neither is minimum weight drawable. We give the proof only for the graph of Figure 2 (a). First observe that for any triangulation, any vertex v of degree 3 having angles 0 ; 1 ; 2 must satisfy m( 0 ) + m( 1 ) + m( 2 ) 2, since each angle is less than . By the condition of Lemma 3.1, at least one of the two variables 0 or must be 0. Without loss of generality, assume that 0 = 0. By the observation beginning this proof, both and 00 must equal 1. This implies, by the condition of Lemma 3.1, that = 0 and 00 = 0. Again, 0 and must equal 1. Repeating this process, we get that = 0 and so 0 = 00 = 1. Similarly, 0 = 0 and so = 00 = 1. Finally, this implies that 0 = 0. But now both and 0 equal 0, contradicting the observation about vertices of degree 3. 2 The triangulations of Figure 2 (a) and (b) are just two examples of in nite classes of minimum weight forbidden triangulations. Other forbidden triangulations can be constructed from these two.
Theorem 3.1 Any triangulation containing either the graph of Figure 2 (a) or the graph of Figure 2 (b) as an induced subgraph is minimum weight forbidden.
Proof: Let T 0 be a triangulation which is minimum weight drawable and let T be an induced subgraph of 6
T 0 such that T is also a triangulation. We will show that the angles of T satisfy the condition of Lemma 3.1; therefore T cannot be either of the graphs of Figure 2. Let f1 and f2 be faces of T that share an edge e, and let v1 and v2 be the vertices of f1 and f2 opposite e. It suces to show that m( (v1 ; f1 ))+ m( (v2 ; f2 )) 1. In T 0 , let f10 and f20 be the faces of T 0 that share edge e, and let v10 and v20 be the vertices of f10 and f20 opposite e. Then, m( (v1 ; f1 )) + m( (v2 ; f2 )) m( (v10 ; f10 )) + m( (v20 ; f20 )) 1; T 0 is minimum weight drawable and each v0 is in the interior of f .
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3.2 Minimum weight drawable and Delaunay forbidden triangulations Lemma 3.3 Every graph in the class described by Theorem 3.1 is also Delaunay forbidden. Sketch of Proof: By removing the vertices drawn as white circles (and their incident edges) in Figure 2 one
can observe that both triangulations violate either Condition 1 (Figure 2 (b)) or Condition 2 (Figure 2 (a)) of Theorem 2.1. 2 The above lemma motivates us to investigate the relationship between Delaunay forbidden and minimum weight forbidden triangulations.
Lemma 3.4 The triangulation of Figure 3 is Delaunay forbidden and minimum weight drawable. Sketch of Proof: By removing the vertices drawn as white circles from Figure 3 (a) one can see that
the triangulation violates Condition 2 of Theorem 2.1, and hence it is not Delaunay drawable. It is not dicult to show that by careful placement of the three o -center interior vertices of Figure 3 (b), we obtain a minimum weight triangulation of the seven points. Let the outer vertices be a0 ; a1 ; a2 , the nearest interior vertex to a be b , and the center vertex be c. Note that all triangulations of the seven points must contain fa0 a1 ; a1 a2 ; a2 a0 g. Moreover, by positioning each b both close enough to a and so that the line a b intersects segment a +2 b +2 , we get that all triangulations of the seven points must also contain each of a b , and a b +1 , for 0 i 2. Since the triangulation in the Figure uses these edges along with the six shortest remaining edges, it must be the minimum weight triangulation of the seven points. 2 The result of Lemma 3.4 can be generalized to an in nite class of triangulations that are Dealunay forbidden and minimum weight drawable. Notice that in the triangulation of Figure 3 (a) there are three subsets of vertices each inducing K4 as a subgraph (see for example the subset fa0 ; c; a1 ; bg. The proof of the following theorem is based on generalizing the reasoning given in the proof of Lemma 3.4. i
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Theorem 3.2 There exists an in nite class of triangulations that are Delaunay forbidden and minimum
weight drawable. The graphs of this class are obtained by k-spining some K4 in the triangulation of Figure 3 (a) for k > 0.
Figure 3 (c) shows an example of a triangulation that belongs to the class described in Theorem 3.2, obtained by 1-spining in Figure 3 (a) the K4 induced the set of vertices fa0 ; c; a1 ; bg.
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Figure 3: (a) A triangulation T that is Delaunay forbidden triangulation, (b) A drawing of T that is a minimum weight triangulation, and (c) A forbidden Delaunay and minimum weight drawable triangulation obtained by 1-spining the subgraph of T induced by the vertices fa0 ; c; a1 ; bg.
4 Classes of Minimum Weight Drawable Triangulations In this section we construct several classes of minimum weight drawable triangulations. The rst two subsections discuss triangulations whose skeleton is either a forest (Subsection 4.1) or a set of nested triangles (Subsection 4.2). Wheel graphs are a special case of the rst class of triangulations. The last subsection (Subsection 4.3) is devoted to k-spined triangulations.
4.1 Wheel Graphs and Skeletons In the previous section we established that a triangulation T may contain a subgraph which prevents T from being drawn as a minimum weight triangulation. This leads us to examine the structure of the skeleton of the triangulation. We show here that any forest can occur as the skeleton of a minimum weight triangulation. This is accomplished by describing methods for
Drawing any fan as a minimum weight triangulation. Extending a minimum weight drawing of a fan by adding an additional fan. Repeating the previous step to build minimum weight drawings the skeletons of which are arbitrary forests.
The simplest fan of interest to us is a kite: a fan with apex a having exactly three neighbors b; c; d. Clearly such a graph is minimum weight drawable; in fact any fan is minimum weight drawable. Figure 4 (a) illustrates a minimum weight drawing of a fan. We will be interested in a certain type of minimum weight drawing of a fan. Let d(x; y) denote the Euclidean distance between points x and y in the plane.
Lemma 4.1 Let F = fa; v1 ; : : : ; v g be a fan with apex a, Consider a drawing of F such that v1 ; : : : ; v are collinear and for all 1 i < j < k n, d(v ; v ) > d(v ; a). Then there exists r > 0 such that if each n
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v is moved away from a along the line v a by a distance of at most r so as to form a convex drawing of F , the drawing so formed is a minimum weight drawing. i
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Sketch of Proof: Let v0 be the new position of v , for 1 i n. It suces to show that d(v0 ; v0 ) > d(v0 ; a) for all 1 i < j < k n, since this will guarantee that the minimum weight triangulation of these points i
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will contain all edges v a. Choose any r > 0 such that r < 13 minfd(v ; v ) ? d(v ; a) : 1 i < j < k ng: Then d(v0 ; v0 ) ? d(v0 ; a) (d(v ; v ) ? 2r) ? (d(v ; a) + r) (d(v ; v ) ? d(v ; a)) ? 3r > 3r ? 3r = 0 2 Using this lemma, it is easy to see that any wheel has a minimum weight drawing which can be obtained as follows: Delete one of the exterior vertices of the wheel, draw the fan that remains as in the preceding lemma, then replace the deleted vertex suitably far away from the rest of the fan. We now describe a method for joining a minimum weight drawing of a given triangulation to one of a fan; it will be of use in our later constructions. Let F be a fan with apex a having neighbors fv1 ; : : : ; v g which has been drawn according to Lemma 4.1. Refer to Figures 4 (a) and (b). Fix a particular vertex v ; 1 < i < n and let ; > 0 be any angles satisfying the following conditions: 1. + is no larger than 6 v ?1 v v +1 ; 2. if i 3 then + 6 v ?2 v ?1 v < ; and 3. if i n ? 2 then + 6 v +2 v +1 v < where all angles are internal angles of F . Let L ?1 be the line through v ?1 making counterclockwise angle with segment v ?1 v , and let L +1 be the line through v +1 making clockwise angle with segment v +1 v . Let R be the quadrilateral region bounded by L ?1 , L +1 , v ?1 v , and v +1 v , and let D (v1 ) be the circle of radius r centered at v1 , where r satis es r < 21 minfd(v ; v ) ? d(v ; a) : 1 i < j < k ng: Finally, S (r; ; ) = D (v1 ) \ R; S (r; ; ) is called a safe region of F at v . i
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Lemma 4.2 Let F = fa; v1 ; : : : ; v g be a fan with apex a, let S (r; ; ) be a safe region of F at v , 1 < i < n, and let X be a nite subset of S (r; ; ). Then a minimum weight triangulation of P = fa; v1 ; : : : ; v g [ X can be obtained as the union of a minimum weight drawing of F , any minimum weight triangulation of X [fv g, and all edges from v ?1 and v +1 to X [fv g which cross no edge of the triangulation of X [fv g. n
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Sketch of Proof: The proof consists of showing that certain edges either cannot exist in any or must
exist in every minimum weight triangulation of P . We sketch the order in which these are established. No minimum weight triangulation of P contains v v , for k > j + 1. No minimum weight triangulation of P contains v x, for any x 2 X unless j = i 1. No minimum weight triangulation of P contains xa, for any x 2 X . Every minimum weight triangulation of P contains each v v +1, 1 j < n. The above imply that every minimum weight triangulation of P contains F . No minimum weight triangulation of P contains any edge v ?1 x or v +1x that intersects any edge of any minimum weight triangulation of X [ fv g. Thus any minimum weight triangulation of P must contain some minimum weight triangulation of X [ fv g. The only edges which can be added at this point are the edges from v ?1 and v +1 to X [ fv g which cross no edge of any triangulation of X [ fv g, and all such edges must be added. j
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This determines, up to the triangulation of X [fv g, exactly which edges must appear in any triangulation of P . 2 Observe that the preceding lemma can be used on a fan consisting of a single kite; this gives a second method for constructing a wheel having n > 3 spokes: just attach a fan having n ? 3 radial edges to a kite. Note also that given any fan F , we can choose X in the preceding lemma such that the only minimum weight triangulation of X [ fv g is isomorphic to F . Thus we can view Lemma 4.2 as providing a method for gluing any fan F with apex u to another fan F at (non-apex) vertex v by identifying u with v. The ideas contained in the proof of the lemma can also be used to design an ecient algorithm for producing a minimum weight drawing of a wheel. Since constructing a minimum weight drawing of a wheel with three spokes is trivial, it is straightforward to establish the following result. i
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Theorem 4.1 Every wheel is minimum weight drawable, and its minimum weight drawing can be computed
in linear time in the real RAM model of computation.
In order to prove that every forest can occur as the skeleton of a minimum weight triangulation, we require a method of attaching several triangulations to an existing fan. The preceding lemma, along with the next de nition, will give us one.
De nition 4.1 Let L be a set of line segments in the plane. Two regions A and B of the plane are mutually invisible with respect to L if for each pair of points a 2 A and b 2 B , segment pq intersects some segment of L.
Lemma 4.3 Let F = fa; v1 ; : : : ; v g be a fan with apex a, let F 0 F be a set of pairwise non-adjacent neighbors of a such that v1 ; v 62 F 0 , and for each v 2 F 0 , consider the kite fv ?1 ; v ; v +1 ; ag in F . Choose r > 0 as in Lemma 4.2, and, for each v 2 F 0, choose f ; g so that the safe regions S (r; ; ) are mutually invisible with respect to F . Finally, for each v 2 F 0 , let X be a nite subset of S (r; ; ). Then a minimum weight triangulation of P = fa; v1 ; : : : ; v g [ fX : v 2 F 0 g can be obtained as the union of a minimum weight drawing of F , any minimum weight triangulations of the sets X [ v , for each v 2 F 0 , and all edges from each v ?1 and v +1 to X [ v which cross no edge of the triangulation of X [ v . Sketch of Proof: We begin with a minimum weight drawing of F as in Lemma 4.1. Note that safe regions S (r; ; ) can be chosen so as to be mutually invisible with respect to F by choosing < ? 6 v ?2 v ?1 v and < ? 6 v +2 v +1 v . We then divide the minimum weight drawing of F into face-disjoint sub-fans n
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F such that each F contains exactly one vertex of F 0 . We now apply Lemma 4.2 to each F and X , obtaining minimum weight triangulations of each P = F [ X . This yields a triangulation of P . It suces to show that no minimum weight triangulation of P can contain any edges of the following three types: v v , for k > j + 1; v x , for x 2 X ; or x x , for x 2 X ; x 2 X . This implies that a minimum weight triangulation of P must consist of minimum weight triangulations of each P . 2 i
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Theorem 4.2 Any forest can be realized as the skeleton of some minimum weight triangulation. Sketch of Proof: Consider the family T of triangulations de ned as follows: 1. Every fan is in T . 2. If F is a fan with apex a, F 0 = fu1 ; : : : ; u g F is a set of non-adjacent neighbors of a, and fT1 ; : : : ; T g T , then the triangulation obtained by attaching each T to F at u according to Lemma 4.3 is in T . k
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Lemma 4.3 guarantees that every graph in T is minimum weight drawable. The skeleton of every element of T is a forest and every forest occurs as an element of some (in fact, in nitely many) elements of T . 2
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Triangulations
Lemma 4.4 Let T be a triangulation with triangular outer face, and let T 0 be the triangulation obtained
by deleting the vertices on the outer face of T . If T 0 has a triangular outer face and is minimum weight drawable, so is T .
Sketch of Proof: Let v0, v1, and v2 be the three vertices of the outer face of T . Let ?0 be a minimum
weight drawing of T 0 . We show how to construct a minimum weight drawing ? of T from ?0 by adding vertices v0 , v1 , and v2 and corresponding incident edges. There are two cases two consider: Case 1: Each v (i = 0; 1; 2) is adjacent to two vertices of the outer face of T . Case 2: One of the v is adjacent to all three vertices of the outer face of T . We sketch the proof only for Case 1; the proof of Case 2 is similar. Let S be the set of vertices of 0 ? , and let be the sum of the lengths of the edges in ?0 . Draw a disk D containing ?0 and such that the distance from any of the outer vertices of ?0 to the boundary of D is larger than . Represent v0 , v1 , and v2 as three points p0 , p1 , and p2 on the boundary of D so that the triangle they de ne contains ?0 . It is easy to see that by appropriate positioning of the p on D, a minimum weight triangulation ? of S [ fp0 ; p1 ; p2 g is obtained by adding to ?0 two edges from each p to the vertices of ?0 corresponding to the neighbors of v in T . 2 An application of Lemma 4.4 is the following. i
i
i
i
i
Theorem 4.3 Every k-nested triangulation is minimum weight drawable, and its minimum weight drawing can be computed in linear time in the real RAM model of computation.
4.3
k-spined
Triangulations
Lemma 4.5 Let T be a triangulation with triangular outer face f0, and let T 0 be obtained by adding a vertex v in the outer face of T and connecting v with all vertices of f0 . If T is minimum weight drawable, so is T 0 .
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Sketch of Proof: Let ? be a minimum weight drawing of T , let S be the set of vertices of ?, and let be
the sum of the lengths of the edges in ?. Represent v as a point p whose distance from any of the vertices of the outer face of ? is larger than , and so that p sees every vertex of the outer face of ?. A minimum weight triangulation ?0 of S [ fpg is obtained by adding to ? three edges, each connecting a vertex of the outer face of ? to p. Clearly, ?0 is a minimum weight drawing of T 0 . 2 An application of Lemma 4.5 is the following.
Theorem 4.4 Every k-spined triangulation is minimum weight drawable, and its minimum weight drawing can be computed in linear time in the real RAM model of computation.
5 Open Problems Several problems remain open towards characterizing which graphs admit a minimum weight drawing. We list here some of those that in our opinion are the most relevant. 1. We have shown in Theorem 4.2 that any forest can be the skeleton of a minimum weight drawable triangulation. Is it the case that every triangulation whose skeleton is a forest is minimum weight drawable? 2. Further investigate the combinatorial relationship between minimum weight and Delaunay drawable triangulations. Theorem 3.2 shows the existence of Delaunay forbidden and minimum weight drawable triangulations. Are there any Delaunay drawable and minimum weight forbidden triangulations? 3. Find other necessary conditions of the type expressed by Lemma 3.1.
References
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