Area Requirement of Gabriel Drawings ?(extended abstract)G. Liotta1, R. Tamassia2, I. G. Tollis3, P. Vocca41 Dipartimento di Informatica e Sistemistica, Universit�a di Roma \La Sapienza", viaSalaria 113, 00198 Rome, Italyliotta@dis.uniroma1.it2 Center for Geometric Computing, Department of Computer Science, BrownUniversity, Providence, RI 02912{1910, USArt@cs.brown.edu3 Department of Computer Science, The University of Texas at Dallas, Richardson,TX 75083{0688, USAtollis@utdallas.edu4 Dipartimento di Matematica, University of Rome \Tor Vergata", via della RicercaScienti�ca, 00133, Rome, Italyvocca@utovrm.itAbstract. In this paper we investigate the area requirement of proxim-ity drawings and we prove an exponential lower bound. Our main contri-bution is to show the existence of a class of Gabriel-drawable graphs thatrequire exponential area for any Gabriel drawing and any resolution rule.The result is further extended to an in�nite class of proximity drawings.1 Introduction.Proximity drawings of graphs have received increasing attention recently in thecomputational geometry and graph drawing communities due to the large num-ber of applications where they play a crucial role. Such applications includepattern recognition and classi�cation, geographic variation analysis, geographicinformation systems, computational geometry, computational morphology, andcomputer vision (see, e.g. [23, 21, 26]).A proximity drawing is a straight{line drawing where two vertices are adjacentif and only if they are neighbors according to some de�nition of neighborhood.? Work supported in part by the US National Science Foundation under grant CCR{9423847, by the US Army Research O�ce under grant DAAH04{96{1{0013, by theNATO Scienti�c A�airs Division under collaborative research grant 911016, by ECESPRIT Long Term Research Project ALCOM-IT under contract no. 20244, byProgetto Finalizzato Trasporti 2 (PFT 2) of the Italian National Research Council(CNR), and by the NATO{CNR Advanced Fellowships Programme. This researchwas done when the �rst author was with the Center for Geometric Computing, De-partment of Computer Science, Brown University,Providence, RI 02912{1910, USA.This research started while the fourth author was visiting the Computer ScienceDepartment of Brown University.

One way of de�ning a neighborhood constraint between a pair of vertices isto use a proximity region, that is a suitable region of the plane having the twopoints on the boundary. Two vertices are adjacent if and only if the correspondingproximity region is empty, i.e., it does not contain any other vertex of the drawing(however, an edge of the drawing may cross the proximity region). For example,two vertices u and v are considered to be neighbors if and only if the closeddisk having u and v as antipodal points, is empty. Proximity drawings that obeythis neighborhood constraint are known in the literature as Gabriel drawings([13, 21]) and the closed disk is calledGabriel disk. A di�erent notion of proximityregion is based on lunes instead of disks. In a relative neighborhood drawing([25, 27]) two vertices u and v are adjacent if and only if the intersection oftwo open disks, one having center at u and the other at v, and with radius thedistance between u and v, is empty. Gabriel drawings and relative neighborhooddrawings are just two examples of an in�nite family of proximity drawings called�{drawings that have been �rst introduced by Kirkpatrick and Radke [16, 23]in the computational morphology context.A �{drawing is a straight{line drawing such that there is an edge between apair of vertices u and v if and only if the corresponding �{lune is empty. The�{lune is de�ned as the intersection of two disks whose radius depends on thevalue of the parameter �. For � � 1, the �{lune is the intersection of the twodisks of radius �d(u; v)=2, where d(u; v) is the distance between u and v, andcentered at the points (1��=2)u+(�=2)v and (�=2)u+(1��=2)v. In particular,for � = 1, the �{lune coincides with the Gabriel disk. Figure 1 depicts a set of�{lunes.u v

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Fig. 1. A set of �{lunes between vertices u and v.

A di�erent de�nition of proximity drawing is given in [8]. A weak proximitydrawing is a straight{line drawing such that if there is an edge between a pairof vertices u, v then the proximity region of u and v is empty. This de�nitionrelaxes the requirement of classical �-drawings, that for each pair of non-adjacentvertices the �-lune is not empty. In other words, if (u; v) is not an edge, thenno requirement is placed on the proximity region of u and v. Several papershave been recently published that characterize proximity drawings and showalgorithms to construct proximity drawings of di�erent classes of graphs anddi�erent de�nitions of proximity [2, 20, 12, 21, 10].In [1, 2], the problem of characterizing �{drawable trees has been addressedand an algorithm to compute Gabriel drawings and relative neighborhood draw-ings of trees in the plane is given. The 3{dimensional version of the same problemhas been studied in [19]. Lubiw and Sleumer [20] proved that maximal outerpla-nar graphs are both relative neighborhood and Gabriel drawings. This result hasbeen extended in [18] to all biconnected outerplanar graphs. Also, in [8] severalalgorithms to construct weak proximity drawings of di�erent families of graphsare given. For a survey on proximity drawings see [7].In this paper, we investigate the area requirement of proximity drawings. The�nite resolution of display and printing devices requires that some constraints beplaced on the drawing so that its dimensions cannot be arbitrarily scaled down.Any constraint which imposes bounds on the minimum distance between ver-tices and (non{incident) edges in the drawing is called a resolution rule. Typicalresolution rules are [4, 9]: The vertex resolution rule which requires that any twovertices have distance at least (a constant) �, (typically, � = 1); the edge reso-lution rule which requires that the minimumdistance between any vertex and anon{incident edge is at least �; the angular resolution rule which states that thevertex resolution rule is veri�ed, and that the minimumangle between two edgesincident at the same vertex is at least �(d), where �(d) is a prede�ned function ofthe maximumdegree of the graph. Once the resolution rule is given, the problemof evaluating the area of a drawing naturally arises [3, 9, 15, 14, 5, 17, 24]. Infact, any resolution rule implies a �nite minimum area for a drawing of a graph.All known algorithms that compute (weak) proximity drawings produce rep-resentations whose area increases exponentially with the number of vertices. Asa consequence, the problem of constructing proximity drawings of graphs thathave small area is considered a very challenging one by several authors [2, 10, 22].Additionally, the importance of this question arises in practice, by observing thatseveral heuristics for drawing graphs often produce proximity drawings [11].We present the �rst results on area requirements of proximity drawings.Namely, we exhibit a class of graphs whose proximity drawings require exponen-tial area under several di�erent de�nitions of proximity. The main contributionsof the paper are listed below.{ We describe a class of planar triangular graphs, such that any Gabriel draw-ing of a graph in the class has area that is exponential in the number ofvertices of the graph. Planar triangular graphs are a classical �eld of studyin graph drawing, because of the many applications where such graphs arise

(see, e.g. [6]).{ We extend the above result to weak Gabriel drawings.{ We show an exponential lower bound on the area of an in�nite class of �{drawings, for 1 � � < 11�cos2�=5 .Some of the proofs are omitted in this extended abstract, because of pagelimits.2 Preliminaries.A graph G = (V;E) consists of a �nite non empty set V (G) of vertices, and a setE(G) of unordered pairs of vertices known as edges. Given an edge e = (u; v),u and v are the endpoints of e and are said to be adjacent vertices. A simplepath of length k in a graph is a �nite sequence P = v1v2 : : : vk, where vi 6= vj ,for 1 � i < j � k, and (vi; vi+1) 2 E(G), for i 2 f1; : : : ; k � 1g. The vertices v1and vk are the endpoints of the path. A k{cycle Ck = v1v2 : : : vk is a sequenceof vertices such that P = v1v2 : : : vk is a simple path and (vk; v1) 2 E(G). Adrawing � of a graph G = (V;E) is a function which maps each vertex of G toa distinct point of the plane and each edge e = (u; v) in G to a simple Jordancurve with endpoints the points of the plane corresponding to u and v. � is astraight{line drawing if each edge is a straight{line segment; � is planar if no twoedges intersect, except possibly at their endpoints. In this paper, when it doesnot give rise to ambiguities, we refer to a drawing of a graph as the graph itself.A planar triangular graph is an embedded planar graph so that every internalface is a 3-cycle, a triangle. The area of a drawing � can be de�ned in severalways depending on whether we evaluate lower or upper bounds. In this paper,we de�ne the area of � as the area of the smallest polygon covering � [9].a

op(a)

1P =Extr(P )2

2P =Intr(P )1Fig. 2. Extruded and Intruded Pentagons.

A Gabriel drawing is a planar straight{line drawing such that there is an edgebetween two vertices u and v if and only if the closed disk having u and v asantipodal points is empty. The closed disk is denoted as D[u; v]. A given graphG is Gabriel drawable if it admits a Gabriel drawing. A maximal Gabriel graph isa Gabriel drawable graph with the maximum number of edges. A weak Gabrieldrawing is a planar straight{line drawing such that if there is an edge betweentwo vertices u and v then the closed disk having u and v as antipodal pointsis empty. A weak Gabriel drawing is contained in or coincides with a Gabrieldrawing.Property1. A Gabriel drawing of a planar triangular graph is such that allinternal faces have acute angles.Property2. [21] In a Gabriel drawing every 3{cycle and every chordless 4{cycle is an internal face.Property3. A weak Gabriel drawing of a maximal Gabriel graph is also aGabriel drawing.In our proofs, we will use several geometric objects. Let IR2 denote the eu-clidean plane. Given any three distinct points a; b; c 2 IR2, 6 abc denotes thecounterclockwise angle between line segments ab and bc; �(abc) denotes thetriangle whose vertices are a, b, and c. Let P1 be a convex pentagon. The inter-sections of the �ve diagonals of P1 de�ne a pentagon P2 inside P1 (see Figure 2).We call P1 the extruded pentagon of P2 and denote it with Extr(P2). Conversely,P2 is the intruded pentagon of P1, denoted with Intr(P1). Let a be a vertex ofP1. Its opposite vertex, op(a), is the vertex of Intr(P1) in the region of the planedelimited by the two diagonals outgoing from a.3 Description of the Class of GraphsOur class of graphs is recursively de�ned as follows. Graph G1 (Figure 3 (a))is a wheel graph with six vertices (the center of the wheel is v�; the othervertices are v10; v11; v12; v13, and v14; the edges are (v�; v1i ) and (v1i ; v1(i+1)mod5), fori 2 f0; : : : ; 4g). For n � 2, Gn is constructed from Gn�1 by adding �ve vertices(denoted as vn0 ; vn1 ; vn2 ; vn3 ; vn4 ), and the edges (vni ; vn(i+1)mod5), (vni ; vn�1i ), and(vni ; vn�1(i+1)mod5), as shown in in Figure 3(b). Fn5 denotes the 5-cycle composedby vn0 vn1 vn2 vn3 vn4 .Property4. Gn is a planar triangular graph, with 5n+1 vertices, and 15n� 5edges. Also, Gn is triconnected.Lemma5. [21] Gn is Gabriel drawable.Figure 4 shows a Gabriel drawing of G2.

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*Fig. 3. A class of graphs that require exponential area.Lemma6. [21] Gn is a maximal Gabriel graph.Lemma7. In a Gabriel drawing of Gn the external face is a 5-cycle.Sketch of Proof. As Gn is triconnected, its planar embedding is �xed for anygiven choice of the external face. From Property 2, the external face of a Gabrieldrawing of Gn can only be the 5{cycle Fn5 = vn0 vn1 vn2 vn3 vn4 .Lemma8. In a Gabriel drawing of Gn (n � 2), for all 1 � i � n � 1, F i5 is astrictly convex pentagon.Sketch of Proof. We prove the lemma by contradiction. Suppose Fn�15 is drawnas a concave pentagon (see Figure 5). Let 6 vn�10 vn�11 vn�12 � �. Since thesum of the external angles of a pentagon is equal to 7�, then there exists atleast one external angle, say 6 vn�11 vn�10 vn�14 , such that 6 vn�11 vn�10 vn�14 � 32�.Hence, at least one of the angles 6 vn�11 vn�10 vn0 , 6 vn0 vn�10 vn4 , and 6 vn4 vn�10 vn�14 isgreater than or equal to �2 . Thus, at least one of the triangles �(vn�11 vn�10 vn0 ),�(vn0 vn�10 vn4 ), �(vn4 vn�10 vn�14 ) has a non acute angle. This contradicts Prop-erty 1.4 Area RequirementIn this section, we prove that any Gabriel drawing of Gn requires exponentialarea. Because of Property 3, the result is immediately extended to weak Gabrieldrawings.Before showing the main result, we need a preliminary technical lemma. Lettwo straight lines in general position (not parallel) l1 and l2 be given; let a and

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Fig. 4. A Gabriel drawing of G2.b two points on l1, and c and d two points on l2, such that a is to the left of band c is to the left of d (see Figure 6). Suppose that the crossing point of l1 andl2 lays to the left of a and c. Let o be the crossing point of the line through aand d and of the line through c and b. Then we have:Lemma9. The area of the triangle �(aoc) is smaller than the area of triangle�(bod).Theorem10. A Gabriel drawing and a weak Gabriel drawing of graph Gn with5n+ 1 vertices require area (3n), under any resolution rule assumption.Sketch of Proof. Let An be the area of a Gabriel drawing �n of Gn and letAn�1 be the area of be a Gabriel drawing �n�1 � �n of Gn�1. We prove thatAn � 3An�1. Since A1 � c, for some constant c depending on the resolutionrule, the theorem follows by induction. For the sake of simplicity, we assume inthe following that the index i 2 f0; : : : ; 4g and all the operations on the indexesare modulo 5.We start by studying how to construct �n from �n�1. We adopt the notationof Figure 7: Bi denotes the region of the plane delimited by the two lines pi andpi+1 through vn�1i and vn�1i+1 , perpendicular to the edge (vn�1i ; vn�1i+1 ); HP (vn�1i )is the the half-plane delimited by the line through edge (vn�1i�1 ; vn�1i ) and notcontaining �n�1; Ci = Bi \HP (vn�1i ) \HP (vn�1i+2 ).

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oFig. 6. Illustration for Lemma 9.Claim 1: In order to construct �n from �n�1, vni must be placed inside Ci.Proof of Claim 1: Suppose, for a contradiction, that vni 62 Ci. Three cases arepossible. (i) vni 62 Bi. Let vni be placed to the right of pi. Then 6 vn�1i+1 vn�1i vni > �2 .Hence, from Property 1, edge (vn�1i+1 ; vni ) cannot be drawn. Similarly, vni can-not be placed to the left of pi+1. (ii) vni 62 HP (vn�1i ). According to the abovecondition, each vertex vni is placed in the corresponding Bi region. Thus, inthis case, vn�1i 2 D[vni ; vni�1]: Hence, edge (vni ; vni�1) cannot be drawn. (iii)vni 62 HP (vn�1i+2 ). Similar to the previous case, vn�1i+1 2 D[vni ; vni+1]. Hence, edge(vni ; vni+1) cannot be drawn.Claim 1 implies that a Gabriel drawing �n of Gn must strictly containExtr(Fn�15 ) (see Figure 8), where Extr(Fn�15 ) is the extruded pentagon of theexternal face Fn�15 of Gn�1.We are now ready to show that Area(Extr(Fn�15 )) is at least three times

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Fig. 7. Construction of �n starting from �n�1.Area(�n�1), which implies that Area(�n) > 3Area(�n�1). The proof proceedsin two steps. In the �rst step we prove that Area(Extr(Fn�15 )) � 2Area(�n�1);in the second step we prove that Area(Extr(Fn�15 )) � 3Area(�n�1). We referto Figure 8: wn0 ; wn1 ; wn2 ; wn3 ; wn4 denote the vertices of Extr(Fn�15 ), such thatvn�1(i+3)mod5 is the opposite vertex of wni ; Pni is the triangle �(vn�1i ; vn�1i+1 ; wni );Wni is the triangle �(vn�1i ; wni�1; wni ).The �rst step of the proof is based on the following claim.Claim 2: Area(S4i=0 Pni ) > Area(�n�1).Proof of Claim 2: Let us consider the line segments having as endpoints ver-tices of Extr(Fn�15 ) and the corresponding opposite vertices of Fn�15 . Thesesegments subdivide each Pni in two triangles. With reference to Figure 9(a), letus consider the two triangles �(COD) and �(AOB). Due to Lemma 8, linesegments ED and EB and points C, D, A, and B satisfy the conditions ofLemma 9, hence, Area(�(COD)) is greater than Area(�(AOB)). A similar rea-soning holds for allPni . It is well known that in any arrangement of lines, every in-ternal region is a simply connected region. Thus, Area(S4i=0Pni ) > Area(�n�1).The second step of the proof is based on the following claim.Claim 3: Area(S4i=0Wni ) > Area(�n�1).Proof of Claim 3: Consider the pentagon Intr(Fn�15 ). We extend the notationto the intruded pentagon, so de�ning the Pn�1i 's and Wn�1i 's regions. FromLemmas 8 and 9, we derive that Area(Wni ) > Area(Wn�1i ) + Area(Pn�1i ) +

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ΓFig. 8. Extruded pentagon of Fn�15 .Area(Wn�1i+1 ). Thus, Area(S4i=0Wni ) > Area(S4i=0 Pn�1i )+2Area(S4i=0Wn�1i ).As S4i=0Wn�1i is covered twice, we can recursively apply the above strategy,to Intr(Intr(Fn�15 )). Because of Claim 2, Area(Extr(Fn�15 )) is at least twiceArea(�n�1). Thus, the recursion ends after a �nite number of steps, provingthat Area(S4i=0Wni ) > Area(�n�1).5 Extensions and Open ProblemsThe result of Theorem 10 can be extended to an in�nite family of (�){drawings.Theorem11. A �-drawing and a weak �-drawing of graph Gn with 5n + 1vertices require area (3n), under any resolution rule assumption, for 1 � � <11�cos 2�=5 .Several problems remain open in this area: (1) Study the area of proximitydrawings using di�erent de�nitions of proximity. For example, we �nd inter-esting to investigate the area of relative neighborhood drawings, and minimumspanning trees. For what concerns minimum spanning trees, note that Monmaand Suri [22] conjectured an exponential lower bound on the area requirement.(2) Motivated by our exponential lower bounds, it is interesting to investigateclasses of graphs that admit a Gabriel drawing (or a weak Gabriel drawing) withpolynomial area.

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Fig. 9. Covering of �n�1.References1. P. Bose, G. Di Battista, W. Lenhart, and G. Liotta. Proximity constraints andrepresentable trees. In R. Tamassia and I. G. Tollis, editors, Graph Drawing(Proc. GD '94), volume 894 of Lecture Notes in Computer Science, pages 340{351. Springer-Verlag, 1995.2. P. Bose, W. Lenhart, and G. Liotta. Characterizing proximity trees. Algorithmica,16:83{110, 1996. (special issue on Graph Drawing, edited by G. Di Battista andR. Tamassia, to appear).3. R. P. Brent and H. T. Kung. On the area of binary tree layouts. Inform. Process.Lett., 11:521{534, 1980.4. Marek Chrobak, Michael T. Goodrich, and Roberto Tamassia. Convex drawings ofgraphs in two and three dimensions. In Proc. 12th Annu. ACM Sympos. Comput.Geom., pages 319{328, 1996.5. P. Crescenzi, G. Di Battista, and A. Piperno. A note on optimal area algorithmsfor upward drawings of binary trees. Comput. Geom. Theory Appl., 2:187{200,1992.6. G. Di Battista, P. Eades, R. Tamassia, and I. G. Tollis. Algorithms for drawinggraphs: an annotated bibliography. Comput. Geom. Theory Appl., 4:235{282, 1994.7. G. Di Battista, W. Lenhart, and G. Liotta. Proximity drawability: a survey. InR. Tamassia and I. G. Tollis, editors, Graph Drawing (Proc. GD '94), volume 894of Lecture Notes in Computer Science, pages 328{339. Springer-Verlag, 1995.8. G. Di Battista, G. Liotta, and S. H. Whitesides. The strength of weak proximity.In F. J. Brandenburg, editor, Graph Drawing (Proc. GD '95), volume 1027 ofLecture Notes in Computer Science, pages 178{189. Springer-Verlag, 1996.9. G. Di Battista, R. Tamassia, and I. G. Tollis. Area requirement and symmetrydisplay of planar upward drawings. Discrete Comput. Geom., 7:381{401, 1992.

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