Simultaneous Embedding of a Planar Graph and Its Dual on the Grid

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1 Simultaneous Embedding of a Planar Graph and Its Dual on the Grid Cesim Erten and Stephen G. Kobourov Department of Computer Siene University of Arizona {esim,kobourov}@s.arizona.edu Abstrat. Traditional representations of graphs and their duals suggest the requirement that the dual verties should be plaed inside their orresponding primal faes, and the edges of the dual graph should ross only their orresponding primal edges. We onsider the problem of simultaneously embedding a planar graph and its dual on a small integer grid suh that the edges are drawn as straight-line segments and the only rossings are between primal-dual pairs of edges. We provide an O(n) time algorithm that simultaneously embeds a 3-onneted planar graph anditsdualona(2n 2) (2n 2) integer grid, where n is the total number of verties in the graph and its dual. Key Words. Graph drawing, planar embedding, simultaneous embedding, onvex planar drawing. Introdution In this paper we address the problem of simultaneously drawing a planar graph and its dual on a small integer grid. The planar dual of an embedded planar graph G is the graph G formed by plaing a vertex inside eah fae of G, and onneting those verties of G whose orresponding faes in G share an edge. Eah vertex in G has a orresponding primal fae and eah edge in G has a orresponding primal edge in the original graph G. The traditional manual representations of a graph and its dual, suggest two natural requirements. One requirement is that we plae a dual vertex inside its orresponding primal fae and the other is that we draw a dual edge so that it only rosses its orresponding primal edge. We provide a linear-time algorithm that simultaneously draws a planar graph and its dual using straight-line segments on the integer grid while satisfying these two requirements.. Related Work Straight-line embedding a planar graph G on the grid, i.e., mapping the verties of G on a small integer grid suh that eah edge an be drawn as a straight-line segment and that no rossings between edges are reated, is a well-studied graph drawing problem. The first solution to this problem was given by de Fraysseix, A full version of this extended abstrat is at

2 Pah and Pollak [6] who provide an algorithm that embeds a planar graph on n verties on the (2n 4) (n 2) integer grid. Later, Shnyder [3] present another method that requires grid size (n 2) (n 2). Also, several restritions of this problem have been onsidered. Harel and Sardas [7] provide an algorithm to embed a bionneted graph on the (2n 4) (n 2) grid without triangulating the graph initially. The algorithm of Chrobak and Kant [4] embeds a 3-onneted planar graph on a (n 2) (n 2) grid so that eah fae is onvex. Miura, Nakano, and Nishizeki [] further restrit the graphs under onsideration to 4-onneted planar graphs with at least 4 verties on the outer fae and present an algorithm for straight-line embedding of suh graphs on a ( n/2 ) ( n/2 ) grid. In a paper dating bak to 963, Tutte [4] shows that there exists a simultaneous straight-line representation of any planar graph and its dual in whih the only intersetions are between orresponding primal-dual edges. However, a disadvantage of this representation is that the area required by the algorithm an be exponential in the number of verties of the graph. Brightwell and Sheinerman [2] show that every 3-onneted planar graph G an be represented as a olletion of irles, one irle representing eah vertex and eah fae, so that, for eah edge of G, the four irles representing the two endpoints and the two neighboring faes meet at a point. Moreover, the vertexirles ross the fae-irles at right angles. This result implies that one an represent a 3-onneted planar graph and its dual simultaneously in the plane with straight-line edges so that the primal edges ross the dual edges at right angles (provided that the vertex orresponding to the unbounded fae is loated at infinity). Mohar [2] extends the results of [2] by presenting an approximation algorithm that given a 3-onneted planar graph G = (V,E) and a rational number ɛ>0 finds an ɛ-approximation for the radii and the oordinates of the enters for the primal-dual irle representation for G and its dual. Mohar s algorithm runs in time polynomial in E(G) and log(/ɛ) and the angles of the primal-dual edge rossings are arbitrarily lose to π/2. Bern and Gilbert [] address a variation of the simultaneous planar-dual embedding problem: finding suitable loations for dual verties, given a straightline planar embedding of a planar graph, so that the edges of the dual graph are also straight-line segments and ross only their orresponding primal edges. They present a linear time algorithm for the problem in the ase of onvex 4- sided faes and show that the problem is NP-hard for the ase of onvex 5-sided faes..2 Our Results The simultaneous embedding in [2] guarantees right angles for the primal-dual edge rossings where the unbounded fae needs to be handled in a speial way by reating a vertex at infinity. Even without onsidering the unbounded fae, the methods in [2] and [2] do not provide bounds on the area required for the simultaneous embedding and they are less pratial than our approah. In this paper we present an algorithm for embedding a given planar graph G and its dual simultaneously so that following onditions are met:

3 Fig.. If the vertex orresponding to the outerfae is drawn expliitly, then one of the edges emanating from it must have a bend. The primal graph is drawn with straight-line segments without rossings. The dual graph is drawn with straight-line segments without rossings. Eah dual vertex lies inside its primal fae. A pair of edges ross if and only if the edges are a primal-dual pair. Both the primal and the dual verties are on the (2n 2) (2n 2) grid, where n is the number of verties in the primal and dual graphs. The running time of the algorithm is O(n). Similar to most primal-dual representation approahes, the unbounded (outer) fae must be treated differently. If the vertex orresponding to the unbounded fae is not expliitly drawn in the plane, then all of the onditions above are met. However, if it is drawn expliitly, then one of the dual edges emanating from it annot be a straight-line segment; see Fig.. In our grid-embedding algorithm, we provide an option for not drawing the vertex representing the outer fae expliitly, or if it is drawn, then one of the edges emanating from it has one bend (that also is on the grid). Note, that if the embedding is done on the surfae of a sphere, the edges emanating from this vertex are ars of great irles and the unbounded fae does not require speial treatment. In setion 2 we desribe the algorithm in detail and in setion 3, we briefly disuss the implementation and present several drawings of primal-dual graphs produed by our algorithm. 2 Algorithm for Embedding a Graph and Its Dual Let G be a 3-onneted planar graph. We onstrut a new graph G 2 that ombines information about both the planar graph G and its dual. For this onstrution we make some hanges in G. We introdue a new vertex v i orresponding toafaef i of G, for all i f, wherefis the number of faes of G.We onnet eah newly added vertex v i to eah vertex v j of F i with a single new edge and delete all the edges that originally belonged to G.Fig.2showsasample onstrution. We all the resulting planar graph G 2 fully-quadrilateralated (FQ), i.e., every fae of G 2 is a quadrilateral. Sine the original graph G is 3- onneted, the resulting graph G 2 is also 3-onneted (proven formally in [4]).

4 0 w v u Fig. 2. Creating graph G 2: the original 3-onneted graph G is drawn with solid lines and filled-in irles; we insert the dual verties (drawn as empty irles) and add the edges onneting primal and dual verties (drawn as dashed lines). To obtain G 2 we remove the original edges of G (drawn with solid lines). w Observation: If we an embed the graph G 2 on the grid so that eah inner fae of G 2 is stritly onvex and the outer fae of G 2 lies on a stritly onave quadrilateral, then we an embed the initial graph G and its dual so that we meet all the problem requirements with the only exeption that one edge of the primal graph G (or its dual) is drawn with one bend. The requirement that the edges of the dual graph be straight and ross only their orresponding primal edges is guaranteed by the strit onvexity of the quadrilateral faes. Let the outer fae of the graph G 2 be (u, v, w, w ), where u, w are primal verties and v, w are dual verties, as shown in Fig. 2. The exeption arises from the fat that we need to draw (u, w) and(v, w ), while both of these edges an not lie inside the quadrilateral (u, v, w, w ). In order to get around this problem we embed the quadrilateral (u, v, w, w )sothatitis stritly onave. This way only one bend for one of the edges (u, w) or(v, w ) will be suffiient. As a result all the edges in the primal and the dual graph are straight-line edges, exept for one edge. In fat, it is easy to hoose the exat edge we need (either from the primal or from the dual). Hene, the original problem an be transformed into a problem of straight-line embedding an FQ-3-onneted planar graph G on the grid so that eah internal fae of G is stritly onvex and the outer fae of G lies on a stritly onave quadrilateral. Note that this problem an be solved by the algorithm of Chrobak et al. [3]. However, the area guaranteed by their algorithm is O(n 3 ) O(n 3 ), whereas our algorithm guarantees a drawing on the (2n 2) (2n 2) grid, whih is stated in the main theorem in this paper: Theorem. Given a 3-onneted planar graph G, we an embed G and its dual on a (2n 2) (2n 2) grid, where n is the number of verties in G and its dual, so that eah dual vertex lies inside its primal fae, eah dual edge rosses

5 only its primal edge and every edge in the overall embedding is a straight-line segment exept for one edge whih has a bend plaed on the grid. Furthermore, the running time of the algorithm is O(n). 2. Overview of the Algorithm Given a 3-onneted graph G, we summarize our algorithm to simultaneously embed G and its dual as follows: Find a topologial embedding of G using [8]. Apply the onstrution desribed above to find G 2. Let G = G 2,whereGis an FQ-3-onneted planar graph. Find a suitable anonial labeling of the verties of G. Plae the verties of G on the grid one at a time using this ordering. Remove all the edges of G and draw the edges of G and its dual. Note that our method works only for 3-onneted graphs. A ommonly used tehnique for drawing a general planar graph is to embed the graph after fully triangulating it by adding some extra edges and then to remove the extra edges from the final embedding. Using the same idea, we ould first fully triangulate any given planar graph. Then after embedding the resulting 3-onneted planar graph and its dual, we ould remove the extra edges that were inserted initially. However, the problem with this approah is that after removing the extra edges there ould be faes with multiple dual verties inside. Thus the issue of hoosing a suitable loation for the duals of suh faes remains unresolved. In fat, depending on the drawing of that fae, it ould as well be the ase that no suitable loation for the dual exists []. In the rest of the paper we onsider only 3-onneted graphs. 2.2 The Canonial Labeling We present the anonial labeling for the type of graphs under onsideration. It is a simple restrition of the anonial labeling of [9], whih in turn is based on the ordering defined in [6]. Let G be an FQ-3-onneted planar graph with n verties. Let (u, v, w, w ) be the outer fae of G s.t. u, w are primal verties and v, w are dual verties. Then there exists a mapping δ from the verties of G onto v i, i msuh that δ maps u and v to v, w to v m and satisfies the following invariants for every 3 k m:. The subgraph G k G, indued by the verties labeled v i, i k is bionneted and the boundary of its exterior fae is a yle C k ontaining the edge (u, v). 2. Either one vertex or two verties an be labeled v k. (a) Let z 0 be the only vertex labeled v k.thenz 0 belongs to the exterior fae of G k, has at least two neighbors in G k and at least one neighbor in G G k.

6 0 v k =z G k 0000 u v z 0 z G 0 0 k u v (a) (b) Fig. 3. (a) One vertex, z 0, is labeled v k ; (b) Two verties, z 0 and z, are labeled v k. (b) Let z 0,z be the two verties labeled v k,where(z 0,z )isanedgein G.Thenz 0,z belong to the outer fae of G k, eah has exatly one neighbor in G k and at least one neighbor in G G k. Sine G is FQ, all the faes reated by adding v k,3 k m,havetobe quadrilaterals; see Fig. 3. Note that assigning the mappings onto v and v m as above provides us the embedding where all the edges of both the primal and the dual graph are straight exept for one primal edge, (u, w), whih has a bend. Alternatively assigning v and w to map onto v,anduto map onto v m would hoose a dual edge, (v, w ), to have a bend. Lemma. Every FQ-3-onneted planar graph has a anonial labeling as defined above. Kant [9] provides a linear-time algorithm to find a anonial labeling of a general 3-onneted planar graph. It is easy to see that the anonial labeling definition of [9] when applied to FQ-3-onneted planar graphs, gives us the labeling defined above. 2.3 The Plaement of the Verties The main idea behind most of the straight-line grid embedding algorithms is to ome up with a suitable ordering of the verties and then plae the verties one at a time using the given order, while making sure that the newly plaed vertex (or verties) is (are) visible to all the neighbors. In order to realize this last goal, at eah step, a set of verties are shifted to the right without affeting the planarity of the drawing so far. Our plaement algorithm is similar to the algorithm of Chrobak and Kant [4], with some hanges in the invariants that we maintain to guarantee the visibility together with strit onvexity of the faes.

7 Let the anonial labeling, δ, that maps the verties of G onto v,v 2,...v m be defined as in the previous setion. Let U(g i ) denote the verties under g i. U(g i ) should be shifted to the right whenever the vertex g i is shifted to the right. U(g i ) is initialized to {g i } for every vertex g i of G. Let δ(g i )=v i and δ(g j )=v j. Then we define Low(g i,g j )=iif i <j,low(g i,g j )=jif j <i.ifi =j then let Low(g i,g j ) be the one that is plaed to the left. Let x(g i ), y(g i ) respetively denote the x and y oordinates of the vertex g i. Embed the First Quadrilateral Fae: We start by plaing the verties mapped onto v and v 2. The ones that are mapped onto v are u and v. We plae u at (0, 0) and v at (3, 0). Note that two verties should be mapped to v 2. We plae the vertex that is mapped to v 2 and that has an edge with u at (, ) and the other at (2, ). Then, for every k, 3 k m, we do the following: Update U(g i ): Let C k = (u =, 2,..., r = v). Let p, q C k, respetively be the first and the last neighbor of the vertex(verties) mapped to v k. If only one vertex, z 0,ismappedtov k,weupdateu( p ), U( q )andu(z 0 )as follows: Low( p, p+ )=p+= U( p )=U( p ) U( p+ ) Low( q 2, q )=q 2= U( q )=U( q ) U( q ) U(z 0 )=U(z 0 ) Low( q 2, q ) i=low( p, p+)+ U( i ) We do not hange U(g i ) if two verties, z 0 and z, are mapped to v k. Shift to the right: We then perform the neessary shifting. We shift eah vertex g i r i=q U( i) to the right by one if only one vertex is mapped to v k,by two otherwise. Loate the New Verties: Finally we loate the vertex(verties) mapped to v k on the grid. Let v k denote the number of verties mapped to v k.thenwe have: If p has no neighbors in G G k x(z 0 )=x( p ) y(z 0 )=y( q )+x( q ) x( p ) v k + otherwise x(z 0 )=x( p )+ y(z 0 )=y( q )+x( q ) x( p ) v k If v k = 2 define z also: x(z )=x(z 0 )+ y(z )=y(z 0 ) Upto this step the algorithm is just a restrition of the one in [4] and it guarantees the onvex drawing of the faes. Then, in order to guarantee stritonvexity, we note the following degenerate ases; see Fig. 4:

8 Degeneraies: We hek for the following: If only one vertex, z 0,ismappedtov k (d ) If x(z 0 )=x( p+ )=x( p+2 ) Shift eah vertex g i r i=p+ U( i) to the right by one. Perform the loation alulation for z 0 again. (d 2) If k<mand z 0, q, q+ are aligned and q has no neighbors in G G k Shift eah vertex g i r i=q+ U( i) to the right by one. If two verties, z 0 and z are mapped to v k (d 3) If y(z 0 )=y(z )=y( p ) Shift eah vertex g i r i=q U( i) to the right by one. Perform the loation alulation for z 0 and z again. (d 4) If k<mand z, q, q+ are aligned and q has no neighbors in G G k Shift eah vertex g i r i=q+ U( i) to the right by one. p z 0... p+2 p+ q p... z 0 o 35 q o 35 q+ p z 0 z o 35 q z 0 p z o 35 q o 35 q+ (a) (b) () (d) Fig. 4. Four possible degenerate ases: type d,typed 2,typed 3,typed Proof of Corretness Lemma 2. Let C k =(u=, 2,..., r = v) be the exterior fae of G k after the k th plaement step. Let α( j, j+ ) denote the angle of the vetor j j+,for j r. The following holds for 2 k m :. α( j, j+ ) lies in [ 45, artan /2] {0} [45, 90 ].Itannotliein ( 45, artan /2] if j has a neighbor in G G k. 2. If j C k, j / {, r } s.t. j does not have a neighbor in G G k,then: (a) If Low( j, j )=j then α( j, j+ )=90 otherwise α( j, j )= 45. (b) If α( j, j+ )=90 then α( j, j ) 90. () If α( j, j+ )= 45 then α( j, j ) 45. Preserving Planarity Let only one vertex, z 0,bemappedtov k.if(z 0, j ) is an edge in G k for some j C k, then the plaement algorithm and the previous lemma guarantees that 90 < α(z 0, j ) < 45, for j p, j q. Then no rossing is reated between a new edge (z 0, j ) and the edges of C k. Beause suh a rossing would imply that there exists j <js.t. j C k and α( j, j ) < 45. But this is impossible by the first part of the above lemma. The proof of this lemma is in drawing.ps

9 z 0 j 0 j+ j+2 j+2 j+ z 0 j z z 0 p q (a) (b) () Fig. 5. The verties pointed to by the arrows must lie in the indiated area. The dashed lines indiate open boundaries that are not inluded in the area. The same idea applies to the ase where v k = 2. Then the following orollary holds: Corollary. Insertion of the vertex(verties) mapped to v k,atthek th plaement step, where 2 k m preserves planarity. Stritly Convex Faes Let v k =andz 0 be the vertex mapped to v k. Let F j =( j, j+, j+2,z 0 ) be a quadrilateral fae reated after the insertion of z 0.IfLow( j, j+ )=j+, then by the previous lemma α( j, j+ )= 45. Fig. 5(a) shows the area where z 0 and j+2 must lie. If Low( j, j+ )=j,then α( j+, j+2 )=90. Fig. 5(b) shows the area where z 0 and j+2 must lie in this ase. Both ases imply that F j =( j, j+, j+2,z 0 ) is stritly onvex. If v k =2andz 0,z are mapped to v k, the plaement algorithm requires that p must lie in the area shown in Fig. 5(), whih implies that the newly reated fae is stritly onvex. The following orollary holds: Corollary 2. The newly reated faes after the insertion of the vertex(verties) mapped to v k,atthek th plaement step, where 2 k m, are stritly onvex. Shifting Preserves Planarity and Stritly Convex Faes The above disussion shows that after the insertion of the vertex(verties) at the kth plaement step, no new edge rossing is reated and all the newly added faes are stritly onvex. In order to omplete the proof of orretness we only need to prove that the same holds for shifting also: Lemma 3. Let C k =(u=, 2,..., r = v) be the exterior fae of G k after the k th plaement step, where 2 k<m. For any given j, where j r, shifting the verties in r i=j U( i), totherightbysunits preserves the planarity and the stritly onvex faes of G k. Proof Sketh: The laim holds trivially for k = 2. Assume it holds for k = k, where 2 k <m. We assume v k =. The ase where v k = 2 is similar. Let z 0 be the vertex mapped to v k and p, q C k, respetively be the first and the last neighbor of z 0 in G k. If j p then by the indutive assumption the planarity of G k and the stritly onvex faes of G k are preserved. The faes introdued by z 0 shifts

10 rigidly to the right, whih, by the previous orollaries, implies that G k is planar and all its faes are stritly onvex. If j>q, then by the indutive assumption the planarity of G k and the stritly onvex faes are preserved. Sine neither z 0 nor any of its neighbors in G k are shifted the lemma follows. If shifting the newly inserted vertex z 0, we indutively apply the shifting to j = Low( p, p+ )+ing k. By the indutive assumption the planarity and stritly onvex faes are preserved for G k. Sine we applied a shifting starting with j then, all the faes exept the first one are shifted rigidly to the right, whih implies that those faes are stritly onvex. Then the only problem ould arise with the leftmost fae. If Low( p, p+ )=p,then p+, p+2 and z 0 are all shifted to the right by the same amount. Sine initially the fae ( p, p+, p+2,z 0 ) was stritly onvex, it ontinues to be so after shifting those three verties also. In the ase where Low( p, p+ )=p+, the only shifted verties are z 0 and p+2. Again shifting those two verties does not hange the property that the fae is onvex. If j = q, the situation is very similar to the previous ase, exept now the only deformed fae is the rightmost fae, instead of the leftmost one. The same idea applies to this ase also, i.e., given that initially the fae is stritly onvex, it remains to be so after shifting 2.5 Grid Size Lemma 4. The plaement algorithm requires a grid of size at most (2n 4) (2n 4). Proof Sketh: If no degeneraies are reated then the exat grid size required is (n ) (n ). We show that eah degenerate ase an be assoiated with a newly added quadrilateral fae of G. Degenerate ase of type d is assoiated with the fae ( p, p+, p+2,z 0 ). Degenerate ase of type d 2 at some step k of the algorithm, is assoiated with afae(z 0, q, q+,g i ), where g i is a vertex that will be added at some step k >kof the algorithm. We know that suh a fae exists, sine k<m, q has no neighbors in G G k and eah fae under onsideration is a quadrilateral. Similar argument holds for degenerate ase of type d 4. Finally degenerate ase of type d 3 is assoiated with the fae ( p, q,z,z 0 ). Fig. 4 shows all four types of degeneraies that an our. Note that eah quadrilateral fae is assoiated with at most one degeneray. Sine an FQ graph G with n verties has n 3 inside faes, the plaement algorithm requires grid size of at most (2n 4) (2n 4). Final Shifting Let (u, v, w, w ) be the outer fae of G. The plaement algorithm and Lemma-2 imply that the outer fae is the isoseles right triangle uvw and that w lies on the line segment (v, w ). We need to do one final right shift to guarantee that the outer fae (u, v, w, w ) lies on a stritly onave quadrilateral. For this we just shift v to the right by one. As a result we an draw

11 Fig. 6. Dodeahedral graph and its dual representation. The filled-in verties and solid edges represent the verties and edges of the primal graph; the empty irles and dashed edges represent verties of the dual graph. the edge (v, w ) as a straight-line segment. In order to draw the edge (u, w), we plae a bend point at (x(w ),y(w ) + 2), where x(w )andy(w ), respetively denote the x and y oordinates of the vertex w. We onnet the bend point with u and w. Then the total area required is (2n 2) (2n 2) and Theorem- follows. 3 Implementation We have implemented our algorithm to visualize 3-onneted planar graphs and their duals using the LEDA/AGD libraries [0]. Finding a suitable anonial labeling takes linear time [9]. We make use of the tehnique introdued by [5] to do the plaement step. It is based on the fat that storing relative x-oordinates of the previously embedded verties is suffiient at every step. Then the plaement step also requires only linear time. Overall, the algorithm runs in linear time. Fig. 6 shows the primal/dual drawing we get for the dodeahedral graph and Fig. 7 shows the primal/dual drawing of an arbitrary 3-onneted planar graph. 4 Aknowledgements We would like to thank Anna Lubiw for introduing us to the problem of simultaneous graph embedding and for stimulating disussions about it. Referenes. M. Bern and J. R. Gilbert. Drawing the planar dual. Information Proessing Letters, 43():7 3, Aug. 992.

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