M. Badent 1, E. Di Giacomo 2, G. Liotta 2

Transcription

1 DIEI Dipartimento di Ingegneria Eettronica e de informazione RT Drawing Coored Graphs on Coored Points M. Badent 1, E. Di Giacomo 2, G. Liotta 2 1 University of Konstanz 2 Università di Perugia 5 December 2006 Via G. Duranti 93, Perugia, Te.: , Fax: , Web: E-mai: segreteria@diei.unipg.it

2 Drawing Coored Graphs on Coored Points Meanie Badent Emiio Di Giacomo Giuseppe Liotta Abstract Let G be a panar graph with n vertices whose vertex set is partitioned into subsets V 0,..., V k 1 for some positive integer 1 k n and et S be a set of n distinct points in the pane partitioned into subsets S 0,..., S k 1 with V i = S i (0 i k 1). This paper studies the probem of computing a crossing-free drawing of G such that each vertex of V i is mapped to a distinct point of S i. Lower and upper bounds on the number of bends per edge are proved for any 2 k n. As a specia case, we improve the upper and ower bounds presented in a paper by Pach and Wenger for k = n [Graphs and Combinatorics (2001), 17: ]. 1 Introduction Let G be a panar graph with n vertices whose vertex set is partitioned into subsets V 0,..., V k 1 for some positive integer 1 k n and et S be a set of n distinct points in the pane partitioned into subsets S 0,..., S k 1 with V i = S i (0 i k 1). We say that each index i is a coor, G is a k-coored panar graph, and S is a k-coored set of points compatibe with G. This paper studies the probem of computing a k-coored point-set embedding of G on S, i.e. a crossing-free drawing of G such that each vertex of V i is mapped to a distinct point of S i. Computing k-coored point-set embeddings of k-coored panar graphs has appications in graph drawing, where the semantic constraints for the vertices of a graph G define the pacement that these vertices must have in a readabe visuaization of G (see, e.g., [6, 15, 18]). For exampe, in the context of data base systems design some particuary reevant entities of an ER schema may be required to be drawn in the center and/or aong the boundary of the diagram (see, e.g., [19]); in socia network anaysis, a typica technique to visuaize and navigate arge networks is to group the vertices into custers and to draw the vertices of a same custer cose with each other and reativey far from those of other custers (see, e.g., [5]). A natura way of modeing these types of semantic constraints is to coor a (sub)set of the vertices of the input graph and to specify a set of ocations having the same coor for their pacement in the drawing. As a resut, the probem of computing k-coored point-set embeddings of k-coored panar graphs has received considerabe interest in the computationa geometry and graph drawing communities, where particuar attention has been devoted to the curve compexity of the computed drawings, i.e. the maximum number of bends aong each edge. Namey, reducing the number of bends aong the edges is a fundamenta optimization goa when computing aestheticay peasing drawings of graphs (see, e.g., [6, 15, 18]). Before presenting our resuts, we briefy review the iterature on the subject. Since there is not a unified terminoogy, we sighty rephrase some of the known resuts; in what foows, n denotes both the number of vertices of a k-coored panar graph and the number of points of a k-coored set of points compatibe with the graph. Kaufmann and Wiese [16] study the mono-chromatic version of the probem, that is they focus on 1- coored point-set embeddings. Given a 1-coored panar graph G (i.e. a panar graph G) and a (1-coored) set An extended abstract of this paper wi appear in the proceedings of the 10 th Workshop on Agorithms and Data Structures, WADS Work partiay supported by the MIUR Project MAINSTREAM: Agorithms for massive information structures and data streams Department of Computer and Information Science, University of Konstanz. meanie.badent@uni-konstanz.de Dipartimento di Ingegneria Eettronica e de Informazione, Università degi Studi di Perugia. {digiacomo, iotta}@diei.unipg.it 1

3 S of points in the pane they show how to compute a 1-coored point-set embedding of G on S such that the curve compexity is at most two, which is proved to be worst case optima. Further studies on 1-chromatic point-set embeddings can be found in [3, 4, 10]; these papers are devoted to characterizing which 1-coored panar graphs with n vertices admit 1-coored point-set embeddings of curve compexity zero on any set of n points and to presenting efficient agorithms for the computation of such drawings. 2-coored point-set embeddings are studied in [9] where it is proved that subcasses of outerpanar graphs, incuding paths, cyces, caterpiars, and wreaths a admit a 2-coored point-set embedding on any 2-coored set of points such that the resuting drawing has constant curve compexity. It is aso shown in [9] that there exists a 3-connected 2-coored panar graph G and a 2-coored set of points S such that every 2-coored point-set embedding of G on S has at east one edge requiring Ω(n) bends. These resuts are extended in [7], where an O(n og n)-time agorithm is described to compute a 2-coored point-set embedding with constant curve compexity for every 2-coored outerpanar graph; in the same paper, it is aso proved that for any positive integer h there exists a 3-coored outerpanar graph G and a 3-coored set of points such that any 3-coored point-set embedding of G on S has at east one edge having more than h bends. Characterizations of famiies of 2-coored panar graphs which admit a 2-coored point-set embedding having curve compexity zero on any compatibe 2-coored set of points can be found in [1, 2, 12, 13, 14]. Key references for the n-chromatic version of the probem are the works by Haton [11] and by Pach and Wenger [17]. Haton [11] proves that an n-coored panar graph aways admits an n-coored point-set embedding on any n-coored set of points; however, he does not address the probem of optimizing the curve compexity of the computed drawing. About ten years ater, Pach and Wenger [17] re-visit the question and show that an n-coored panar graph G aways has an n-coored point-set embedding on any n-coored set of points such that each edge of the drawing has at most 120n bends; they aso give a probabiistic argument to prove that, asymptoticay, the upper bound on the curve compexity is tight for a inear number of edges. More precisey, et G be an n-coored panar graph with m independent edges and et S be a set of n points in convex position such that each point is coored at random with one of n distinct coors. Pach and Wenger prove that, amost surey, at east m m 20 edges of G have at east 40 bends on any n-coored point-set embedding 3 of G on S. The present paper describes a unified approach to the probem of computing k-coored point-set embeddings for 2 k n. The research is motivated by the foowing observations: (i) The iterature has either focused on very few coors or on the n coors case; in spite of the practica reevance of the probem, itte seems to be known about how to draw graphs where the vertices are grouped into 2 k n custers and there are semantic constraints for the pacement of these vertices. (ii) The Ω(n) ower bound on the curve compexity for 2-coored point-set embeddings described in [9] impies that for any 2 k n there can be k-coored point-set embeddings which require a inear number of bends per edge. This coud ead to the concusion that in order to compute k-coored point-set embeddings that are optima in terms of curve compexity one can arbitrariy n-coor the input graph, consistenty coor the input set of points, and then use the drawing agorithm by Pach and Wenger [17]. However, the ower bound of [9] shows Ω(n) curve compexity for a constant number of edges, whereas the drawing technique of Pach and Wenger gives rise to a inear number of edges each having a inear number of bends. Hence, the tota number of bends in a drawing obtained by the technique of [17] is O(n 2 ) and it is not known whether there are sma vaues of k for which o(n 2 ) bends woud be aways possibe. (iii) There is a arge gap between the mutipicative constant factors that define the upper and the ower bound of the curve compexity of n-coored point-set embeddings [17]. Since the readabiity of a drawing of a graph is strongy affected by the number of bends aong the edges, it is natura to study whether there exists an agorithm that guarantees curve compexity ess than 120n. Our main resuts are as foows. A ower bound on the curve compexity of k-coored point-set embeddings is presented which estabishes that Ω(n 2 ) bends may be necessary even for sma vaues of k. Namey, it is shown that for every n such that n 16 and for every k such that 2 k n there exists a k-coored panar graph G with n vertices and a k-coored set of points S compatibe with G such that any k-coored point-set embedding of G on S has Ω(n) edges each having Ω(n) bends. This ower bound generaizes the one in [17] for k = n and the one in [9] for k = 2. Aso, the constant factors of our ower bound for k = n are significanty 2

4 arger than those in [17]. An O(n 2 og n)-time agorithm is described that receives as input a k-coored panar graph G (2 k n), a k-coored set of points S compatibe with G, and computes a k-coored point-set embedding of G on S with curve compexity at most 3n + 2. This reduces by about forty times the previousy known upper bound for k = n [17]. Motivated by the previousy described ower bound, specia coorings of the input graph are studied which can guarantee a curve compexity that does not depend on n. Namey, it is shown that if the k-coored panar graph G has k 1 vertices each having a distinct coor and n k + 1 vertices of the same coor, it is aways possibe to compute a k-coored point-set embedding whose curve compexity is at most 9k 1. Both the ower and the upper bounds are proved by using a common technique, based on transating the geometric probem into a topoogica augmentation probem. The upper bounds are based on an agorithm that computes a panar drawing of a graph such that a vertices are coinear, the vertices foow a given eft-to-right order, and the edges rippe ony a few times. The remainder of this paper is organized as foows. Preiminary definitions are in Section 2. The ower bound is described in Section 3. Sections 4, 6, and 7 are devoted to the drawing agorithms and their anaysis both in terms of computationa compexity and in terms of curve compexity. Concusions and open probems can be found in Section 8. 2 Preiminaries A drawing of a graph G is a geometric representation of G such that each vertex is a distinct point of the Eucidean pane and each edge is a simpe Jordan curve connecting the points which represent its endvertices. A drawing is panar if any two edges can ony share the points that represent common endvertices. A graph is panar if it admits a panar drawing. Let G = (V, E) be a graph. A k-cooring of G is a partition {V 0, V 1,..., V k 1 } of V where the integers 0, 1,..., k 1 are caed coors. In the rest of this section the index i is 0 i k 1 if not differenty specified. For each vertex v V i we denote by co(v) the coor i of v. A graph G with a k-cooring is caed a k-coored graph. Let S be a set of distinct points in the pane. We aways assume that the points of S have distinct x-coordinates (this condition can aways be satisfied by means of a suitabe rotation of the pane). For any point p S we denote by x(p) and y(p) the x- and y-coordinates of p, respectivey. A k-cooring of S is a partition {S 0, S 1,..., S k 1 } of S. A set S of distinct points in the pane with a k-cooring is caed a k-coored set of points. For each point p S i co(p) denotes the coor i of p. A k-coored set of points S is compatibe with a k-coored graph G if V i = S i for every i; if G is panar, we say that G has a k-coored point-set embedding on S if there exists a panar drawing of G such that: (i) every vertex v is mapped to a distinct point p of S with co(p) = co(v), (ii) each edge e of G is drawn as a poyine λ; a point shared by any two consecutive segments of λ is caed a bend of e. The curve compexity of a drawing is the maximum number of bends per edge. Throughout the paper n denotes the number of vertices of graph and m the number of its edges. 3 Lower Bounds on the Curve Compexity In this section, we first show that for any integer k such that 3 k n, the probem of computing k-coored point-set embeddings can require a inear number of edges each having a inear number of bends. Then, we show how this resut can be extended to 2-coored point-set embeddings. The ower bound technique for 3 k n is based on a deterministic proof and uses combinatoria arguments. We first describe a 3-coored panar graph with n vertices and a 3-coored set of points compatibe with this graph. We then show a property of any 3-coored point-set embedding of this graph on the set of 3

5 points; we finay describe a topoogica property of the graph. The union of the two properties gives rise to the ower bound. Since the ower bound for the specia case of 2-coored point-set embeddings can be proved by means of the same approach but with sight differences in the constant factors, we just state the resut in this section and refer the interested reader to the paper appendix for a detaied proof. 3.1 Diamond Graphs and 3-coored Sets of Points w 9 w 5 w 1 w 10 w 6 w 2 u 3 u 2 u 1 u 11 u 10 u 9 u 0 u 4 u 5 u 6 u 7 u 8 w 0 w 4 w 8 v 8 v 9 v 10 v 11 v 7 v 6 v 5 v 4 v 0 v 1 v 2 v 3 w 3 w 7 w 11 (a) (b) Figure 1: (a) A diamond graph G n. (b) A 3-coored set of points with an aternating bi-coored sequence compatibe with G n. A diamond graph is a 3-coored panar graph as the one depicted in Figure 1(a). More formay, et n 12, et n = (n mod 12) and et n = n n = 12h for some h > 0; a diamond graph G n = (V, E) is defined as foows: V = V 0 V 1 V 2 V 0 = {v i 0 i n 3 + n 2 } V 1 = {u i 0 i n 3 + V 2 = {w i 0 i n 3 } n 2 } E = E 0 E 1 E 2 E 3 E 4 E 0 = {(v i, v i+1 ) 0 i n 3 + n 2 1} E 1 = {(u i, u i+1 ) 0 i n 3 + n 2 1} E 2 = {(w i, w i+1 ), (w i+1, w i+2 ), (w i+2, w i+3 ), (w i+3, w i ) 0 i 4h 1, i mod 4 = 0} E 3 = {(w i+1, w i+4 ), (w i+3, w i+4 ), (w i+1, w i+6 ), (w i+3, w i+6 ) 0 i 4h 5, i mod 4 = 0} 4

6 E 4 = {(w 4h 1, v n n 3 + ), (w 4h 3, v 0 ), (w 0, u 0 ), (w 2, u n n 2 )} Let S = S 0 S 1 be a 2-coored set of points a beonging to a horizonta straight ine ; we say that S is an aternating bi-coored sequence if S 0 = S 1 or S 0 = S and no two points of the same coor appear consecutivey aong the ine. A 3-coored set of points with an aternating bi-coored sequence is a 3-coored set of points S = S 0 S 1 S 2 such that S = S 0 S 1 is an aternating bi-coored sequence with no point of S 2 on. See Figure 1(b) for an exampe. 3.2 Bi-coored Paths and Lower Bounds Let G n (n 12) be the diamond graph with n vertices and et S be a 3-coored set of points with an aternating bi-coored sequence and compatibe with G n. Let Γ n be a 3-coored point-set embedding of G n on S. In what foows we sha assume that no bend is represented by a point that beongs to the horizonta straight ine that contains the bi-coored sequence of S. Namey, if a point p representing a bend of an edge of Γ n is a point of, we can sighty perturb the drawing so that the drawing remains panar and p is moved either above or beow. Let p 0, p 1,..., p 8h+n 1 be the points of the bi-coored sequence of S ordered according to their x- coordinates. Denote with z i the vertex of G n which is mapped to p i. Notice that z i and z i+1 are not adjacent in Γ n because one of them beongs to V 0 and the other one beongs to V 1 in G n. Connect in Γ n z i and z i+1 with a straight-ine segment (i = 0,..., 8h + n 2); the obtained path is caed bi-coored path on Γ n. Lemma 1 Let G n (n 12) be a diamond graph and et S be a 3-coored set of points with an aternating bi-coored sequence such that S is compatibe with G n. Let Γ n be a 3-coored point-set embedding of G n on S, et e be an edge of Γ n, and et Π be the bi-coored path on Γ n. If Π crosses e b times, then e has at east b 1 bends. Proof: Since no bend of Γ n is on and no vertex of V 2 is on then each segment of e can cross the straight ine that contains the bi-coored sequence of S at most once. Thus, if e is crossed b times by Π, then it consists of at east b segments. Since at most two endpoints of these segments can be the endvertices of e, it foows that e has at east b 1 bends. Lemma 2 Let G n (n 12) be a diamond graph and et S be a 3-coored set of points with an aternating bi-coored sequence such that S is compatibe with G n. Let Γ n be a 3-coored point-set embedding of G n on S and et Π be the bi-coored path on Γ n. Π crosses at east n 6 1 edges of Γ n, where n = n (n mod 12); aso, Π crosses each of these edges at east n 6 times. Proof: For a panar drawing of G n and a cyce C G n we say that C separates a subset V V from a subset V V if a vertices of V ie in the interior of the region bounded by C and a vertices of V are in the exterior of this region. In every panar drawing of G n each of the h cyces defined by the edges in the set E 2 separates a vertices in V 0 from a vertices in V 1. Thus every edge of Π must cross these h cyces. Anaogousy, in every panar drawing of G n each of the h 1 cyces defined by the edges in the set E 3 separates a vertices in V 0 from a vertices in V 1. Therefore, every edge of Π must aso cross these h 1 cyces. The number of edges in Π is 2n 3 + n 1, where n = n n = n mod 12, and hence each cyce is crossed 2n 3 + n 1 times. Since each cyce has four edges, we have that at east 2h 1 = n 6 per cyce) are crossed at east n 6 + n h = 2h 1 4 We are now ready to prove the ower bound. 1 edges (one = 2h = n 6 times. Theorem 1 For every n 12 and for every 3 k n there exists a k-coored panar graph G with n vertices and a k-coored set of points S compatibe with G such that any k-coored point-set embedding of G on S has at east n n 6 1 edges each having at east 6 1 bends, where n = n (n mod 12). 5

7 Proof: Given any n 12 construct a diamond graph G n and consider a 3-coored set of points S with an aternating bi-coored sequence which is compatibe with G n. Arbitrariy divide the set of coors {0, 1,..., k 1} in three non-empty subsets C 0, C 1 and C 2. Arbitrariy coor the vertices of G n in the set V i by using the coors in the set C i (i = 0, 1, 2), with the ony requirement that each coor is used at east once. Anaogousy, arbitrariy coor the points of S in the set S i by using the coors in the set C i (i = 0, 1, 2) with the ony requirement that S remains compatibe with G n. As a resut we have a k-coored graph G n with n vertices and a k-coored set of points S compatibe with G n. Let Γ n be a k-coored point-set embedding of G n on S. Let Π be the bi-coored path on Γ n. By Lemma 2 there are at east n 6 1 edges of Γ n that are crossed by Π at east n 6 times. By Lemma 1 each of these edges has at east n 6 1 bends in Γ n. We can compare the resut of Theorem 1 with the known ower bound for k = n [17]. Let G be an n-coored graph with m independent edges and et S be a set of n points in convex position such that each point is coored at random with one of n distinct coors. In [17] it is proved that, amost surey, at east m 20 edges of G have at east m 40 bends on any possibe n-coored point-set embedding of G on S. A comparison 3 with the resut in Theorem 1 can be easiy done by observing that the maximum number of independent edges in a graph with n vertices is at most n/2. Aso, we remark the argument of Theorem 1 is deterministic and that it can be appied to a vaues of k such that 3 k n. We concude this section by extending Theorem 1 to the case of 2-coored point-set embeddings. The extension uses the same reasoning iustrated above for three or more coors, but it requires sighty different definitions and gives rise to sighty smaer constant factors. Whie a detais have been moved to the paper appendix, we give here ony a brief sketch of the ideas behind this ower bound. Intuitivey, a 2-coored diamond graph can be regarded as a diamond graph where the vertices of set V 1 and V 2 have the same coor and the vertices of set V 0 are such that V 0 = V 1 + V 2. Figure 2(a) is an exampe of a 2-coored diamond graph (see aso the appendix for a forma definition of a 2-coored diamond graph); Figure 2(b) is an aternating bi-coored sequence compatibe with the graph of Figure 2(a). With the same reasoning iustrated above, the foowing can be proved (see the appendix for detais). Theorem 2 For every n 16 there exists a 2-coored panar graph G n with n vertices and a 2-coored set of points S compatibe with G n such that any 2-coored point-set embedding of G n on S has at east n 8 1 edges each having at east n 8 1 bends, where n = n (n mod 16). 4 Upper Bounds: Overview of the Approach Theorems 1 and 2 show that, for every 2 k n, Ω(n) bends per edge can be required in a k-coored pointset embedding of a k-coored graph G with n vertices. Therefore, a drawing agorithm that is asymptoticay optima in terms of curve compexity for a vaues of k such that 2 k n can be designed as foows: (1) Arbitrariy assign each vertex of G having coor i to a distinct point of coor i (if there is more than one vertex of G having coor i); and (2) Appy the drawing agorithm of Pach and Wenger [17], which computes an n-coored point-set embedding of G whose curve compexity is at most 120n. However, since optimizing the number of bends per edge is an important requirement that guarantees the readabiity of a drawing of a graph [6, 15, 18], we present in the next three sections a new drawing strategy that gives rise to n-coored point-set embeddings with curve compexity at most 3n + 2. The key idea is to transate the geometric probem into an equivaent topoogica probem, namey that of computing a Hamitonian path of a panar graph by suitaby augmenting it with dummy edges that do not cross the rea edges too many times. An overview of the content of the next three sections is as foows: The notion of augmenting k-coored Hamitonian path for a k-coored panar graph G is introduced (Section 5). A theorem that proves that the number of crossings between the edges of an augmenting k-coored Hamitonian path and the edges of G define an upper bound on the curve compexity of a k-coored point-set embedding of G is proved (Theorem 3). 6

8 w 9 w 5 w 1 v 16 v 15 v 0 v 17 v 14 v 1 w 10 w 6 w 2 u 3 u 2 u 4 u 5 u 1 u 6 u 0 u 7 w 0 w 4 w 8 v 18 v 13 v 2 v 19 v 12 v 3 u 11 u 10 u 9 u 8 v 20 v 11 v 4 v 21 v 10 v 5 v 22 v 9 v 6 w 3 v 23 v 8 v 7 w 7 w 11 (a) (b) Figure 2: (a) A 2-coored diamond graph G n. (b) An aternating bi-coored sequence compatibe with G n. 7

9 An agorithm that, for any inear ordering of the vertices of G, computes a panar drawing of G such that a vertices are coinear, the vertices in the drawing foow the given ordering, and each edge can be decomposed into at most three x-monotone curves is presented (Section 6). Finay, the above agorithm is expoited to compute a k-coored hamitonian path on G and then a k-coored point-set embedding such that every edge bends at most 3n + 2 times. (Section 7). 5 Coored Hamitonicity A k-coored sequence σ is a inear sequence of (possiby repeated) coors c 0, c 1,..., c n 1 such that 0 c j k 1 (0 j n 1). We say that σ is compatibe with a k-coored graph G if, for every 0 i k 1, coor i occurs V i times in σ. Let S be a k-coored set of points and et p 0, p 1,..., p n 1 be the points of S ordered according to their x-coordinates. We say that S induces the k-coored sequence σ = co(p 0 ), co(p 1 ),..., co(p n 1 ). Figures 3(a) and 3(b) show an exampe of a 3-coored panar graph and of a 3-coored sequence compatibe with it and induced by a 3-coored set of points. A graph G has a Hamitonian path if it has a simpe path that contains a the vertices of G. If G is a k-coored graph and σ = c 0, c 1,..., c n 1 is a k-coored sequence compatibe with G, a k-coored Hamitonian path of G consistent with σ is a Hamitonian path v 0, v 1,..., v n 1 such that co(v i ) = c i (0 i n 1). A k-coored panar graph G can aways be augmented to a (not necessariy panar) k-coored graph G by adding to G a suitabe number of dummy edges and such that G has a k-coored Hamitonian path H consistent with σ and that incudes a dummy edges. Figure 3(c) shows an augmentation of the graph of Figure 3(a) such that the augmented (non-panar) graph has a 3-coored Hamitonian consistent with the sequence of Figure 3(b). If G is not panar, we can appy a panarization agorithm (see, e.g., [6]) to G with the constraint that ony crossings between dummy edges and edges of G H are aowed (see Figure 3(d)). Such a panarization agorithm constructs an embedded panar graph G, caed augmented Hamitonian form of G, where each edge crossing is repaced with a dummy vertex, caed division vertex. By this procedure, an edge e of H can be transformed into a path whose interna vertices are division vertices. The subdivision of H obtained this way is caed an augmenting k-coored Hamitonian path of G consistent with σ and is denoted as H. If every edge e of G is crossed at most d times in G (i.e. e is spit by at most d division vertices in G ), H is said to be an augmenting k-coored Hamitonian path of G consistent with σ and inducing at most d division vertices per edge. Notice that d is the number of division vertices that have been inserted aong each edge of G; for exampe, the path H of Figure 3(d) is an augmenting 3-coored Hamitonian path of the graph of Figure 3(a) consistent with the sequence of Figure 3(b) and inducing one division vertex per edge, because each edge of H crosses each edge of G at most once. If G is panar, then the augmented Hamitonian form of G is G and H coincides with H. If both endvertices of H are on the externa face of the augmented Hamitonian form of G, then H is said to be externa. Let v d be a division vertex for an edge e of G. Since a division vertex corresponds to a crossing between e and an edge of H, there are four edges incident on v d in G ; two of them are dummy edges that beong to H, the other two are two pieces of edge e obtained by spitting e with v d. Let (u, v d ) and (v, v d ) be the atter two edges. We say that v d is a fat division vertex if it is encountered after u and before v whie waking aong H ; v d is a pointy division vertex otherwise. The foowing theorem refines and improves a simiar resut presented in [7]. The agorithm described in its proof is based on the drawing technique of Kaufmann and Wiese [16]. Theorem 3 Let G be a k-coored panar graph with n vertices, et σ be a k-coored sequence compatibe with G, and et H be an augmenting k-coored Hamitonian path of G consistent with σ inducing at most d f fat and d p pointy division vertices per edge. If H is externa then G admits a k-coored point-set embedding on any set of points that induces σ such that the maximum number of bends aong each edge is d f + 2d p + 1. Proof: Let S be a k-coored set of points that induces the k-coored sequence σ. We sha use path H to construct a k-coored point-set embedding of G on S. Let H = w 0, w 1,..., w n 1. Path H contains aso the 8

10 v 10 G v 9 S v 7 v 6 v 8 v 3 v 4 v 5 v 2 σ v 0 v 1 (a) (b) H v 10 H v 10 G v 9 G v 9 v 7 v 6 v 8 v 7 v 6 v 8 v 4 v 3 v 5 v 3 v 4 v 5 v 2 v 2 v 0 v 1 v 0 v 1 (c) (d) Figure 3: (a) A 3-coored panar graph graph G. (b) A 3-coored set of points S consistent with G and its induced 3-coored sequence σ, compatibe with G. (c) An augmentation of G to a (non-panar) 3-coored graph G that admits a 3-coored Hamitonian path H consistent with σ. Path H is highighted in bod. Dashed edges are dummy edges. (d) A panar graph G obtained by appying a panarization agorithm to G. The path highighted in bod is an augmenting 3-coored Hamitonian path H of G consistent with σ and inducing at most 1 division vertex per edge. 9

11 division vertices, which are not vertices of G. We give these vertices a new coor k. In order to draw them we define a new set of points S by adding a suitabe number of points to S, a having coor k and paced so that if q 0, q 1,..., q n 1 are the points of S ordered according to their x-coordinates, then c(q j ) = c(w j ) (j = 0,..., n 1). In the foowing we denote as G the augmented Hamitonian form of G. We can now use the the drawing technique of Kaufmann and Wiese [16] to point set embed G on S ; for competeness, we reca this technique in the foowing. Map each vertex w j to point q j (j = 0,..., n 1) in S and draw the edges of path H as straight-ine segments between their endvertices. Each edge e not in H is drawn by using two segments, one with sope s > 0 and the other with sope s. In order to avoid crossings between e and the edges in H the sope s is chosen to be greater than the absoute vaue of the sope of each edge in H. With segments of sope ±s, it is possibe to draw each edge e above or beow H. Since H is externa, there exists a panar embedding of G such that w 0 and w n 1 are on the externa face. In such an embedding every edge not in H is either on the eft-hand side of H, in which case it is drawn above H, or on the right-hand side of H when waking from w 0 to w n 1, in which case it is drawn beow H. The resuting drawing is panar except that edges outside H that are incident on the same vertex may contain overapping segments. To eiminate overapping, perturb overapping edges by decreasing the absoute vaue of their segment sopes by sighty different amounts. The sope changes are chosen to be sma enough to avoid creating edge crossings whie preserving the same panar embedding. For detais about this rotation see [16]. The drawing obtained by the technique described above is a (k + 1)-coored point-set embedding of G on S with at most one bend per edge. Removing the vertices and edges added to obtain G from G we have a k-coored point-set embedding of G on S. Consider an edge e of G and suppose that e is spit by means of d t = d f + d p division vertices in G. Then there are d t + 1 edges in G corresponding to e, each one having at most one bend. As we pointed out above, there are four edges incident on every dummy vertex d; two of them are dummy edges that beong to H, the other two are two pieces of the rea edge e obtained by spitting e by means of d. After the remova of dummy eements (vertices and edges) ony the atter two edges remain in the drawing. Denote them as (u, d) and (v, d). Since these edges are not in H, one of them is above H and the other one is beow H. Thus a segment s u of (u, d) and a segment s v of (v, d) are incident on d, one from above and one from beow. Since d has ony one segment incident from above and ony one segment incident from beow, the rotation performed to remove overap does not affect s u and s v, which therefore have sope either +s or s. If d is a pointy division vertex then s u and s v have different sopes and the remova of d gives an extra bend; if d is a fat division vertex, then s u and s v have the same sope and d can be removed without introducing any extra bend. Thus we can have d p extra bends for an overa curve compexity of d t d p = d f + 2d p + 1. Based on Theorem 3, we wi show our upper bound by proving that for any n-coored sequence σ an n-coored panar graph G aways admits an augmenting k-coored Hamitonian path of G consistent with σ such that for each edge d f 3n 3 and d p 2. 6 Computing Topoogica Book Embeddings with a Given Linear Ordering The agorithm to compute an augmenting k-coored Hamitonian path of G consistent with σ reies on a geometric technique that starts with a topoogica book embedding of G (a specia type of panar drawing where a vertices are aigned, defined in the next paragraphs) and transforms it into a new topoogica book embedding that respects the given inear ordering for the vertices of G. A spine is an horizonta ine. Let be a spine and et p, q be two points of. An arc is a circuar arc passing through the three points p, q, and r, where r is a point of the perpendicuar bisector of pq, at a distance d(p,q) 4 from. The arc can be either in the haf-pane above the spine or in the haf-pane beow the spine; in the first case we say that the arc is in the top page of, otherwise it is in the bottom page of. Let G = (V, E) be a panar graph. A topoogica book embedding of G is a panar drawing such that a 10

12 vertices of G are represented as points of a spine and each edge can be either above the spine, or beow the spine, or it can cross the spine. Each crossing between an edge and the spine is caed a spine crossing. It is aso assumed that in a topoogica book embedding every edge consists of one or more arcs such that no two consecutive arcs are in the same page. An edge e is said to be in the top (bottom) page of the spine if it consists of exacty one arc and this arc is in the top (bottom) page. Figure 4 shows two exampes of topoogica book embeddings. A monotone topoogica book embedding is a topoogica book embedding such that each edge crosses the spine at most once. Aso, et e = (u, v) be an edge of a monotone topoogica book embedding that crosses the spine at a point p; e is such that if u precedes v in the eft-to-right order aong the spine then p is between u and v, the arc with endpoints u and p is in the bottom page, and the arc with endpoints u and v is in the top page. Figure 4(a) is an exampe of a monotone topoogica book embedding of a panar graph (a) (b) Figure 4: Two topoogica book embeddings of a panar graph G. (a) A monotone topoogica book embedding of G. (b) A 3-chain topoogica book embedding of G. The bod edge consists of three x-monotone chains. Theorem 4 [8] Every panar graph admits a monotone topoogica book embedding. Aso, a monotone topoogica book embedding can be computed in O(n) time, where n is the number of the vertices in the graph. Let e = (u, v) be an edge of a topoogica book embedding. An x-monotone portion of e is a portion π e of e such that every vertica ine intersects π e at most once. An x-monotone portion of e is maxima if it is not contained in any other x-monotone portion of e. A maxima x-monotone portion of e is caed an x-monotone chain of e. We say that a topoogica book embedding is a k-chain topoogica book embedding if each edge consists of at most k x-monotone chains. Figure 4(b) is an exampe of a 3-chain monotone topoogica book embedding of the same graph of Figure 4(a): the bod edge in the drawing consists of three x-monotone chains and a other edges consist of at most two x-monotone chains. Notice that the inear order of the vertices aong the spine in Figure 4(b) is different from the one in Figure 4(a). Before presenting our drawing agorithm to compute a topoogica book embedding with a given eftto-right order of the vertices aong the spine we need to introduce another concept, which generaizes the notion of topoogica book embedding. Let and be two distinct spines such that is above ; is caed upper spine and is caed ower spine. A 2-spine drawing Γ of G is a (not necessariy panar) drawing such that each vertex of G is represented as a point either of the upper spine or of the ower spine and each edge crosses the spines a finite number of times. More precisey, an edge of a 2-spine drawing can have both endvertices in the same spine, or in different spines. If both endvertices are in the same spine, the edge consists of a sequence of arcs such that any two consecutive arcs are on opposite pages of the spine. If one endvertex is in the upper spine and the other is in the ower spine, then the edge consists of: (i) a (possiby empty) sequence of arcs whose endpoints are in the upper spine, caed the upper sequence of the edge; (ii) a straight-ine segment between the two spines, caed the inter-spine segment of the edge; (iii) a (possiby empty) sequence of arcs whose endpoints are in the ower spine, caed the ower sequence of the edge. It is aso assumed that any two consecutive arcs of the upper (ower) sequence are on opposite pages of the upper (ower) spine. In what foows, we sha sometimes treat arcs and inter-spine segments in the same way; in 11

13 these cases we sha use the term sub-edge of an edge to mean either an arc or an inter-spine segment of an edge in a 2-spine drawing. Figure 5 is an exampe of a panar 2-spine drawing of the same graph of Figure 4(a); (1, 3) is an exampe of an edge with both endvertices on the same spine. Edge e = (2, 6) in Figure 5 has its endvertices on different spines: The upper sequence is the sequence of arcs of e from p to 6; the straight-ine segment pq is the inter-spine segment of e; the sequence of arcs of e from q to 2 is the ower sequence of e. Observe that if a vertices are on the same (upper or ower) spine and if the drawing is panar, a 2-spine drawing of a graph is a topoogica book embedding of the graph. p q Figure 5: A 2-spine drawing of the graph in Figure 4(a). segment, and a ower sequence. The bod edge has an upper sequence, an inter-spine 6.1 Agorithm LinearOrderDraw Agorithm LinearOrderDraw receives as input a panar graph G with n vertices and a inear ordering λ of the vertices of G. It produces as output a 3-chain topoogica book embedding Γ of G such that the eft-toright order of the vertices aong the spine of Γ is λ. By using Theorem 4, Agorithm LinearOrderDraw computes first a monotone topoogica book embedding of G, denoted as Γ; then, it transforms Γ into the 3-chain topoogica book embedding Γ. Let be the spine of Γ and et v 0,..., v n 1 be the vertices of G in the eft-to-right order they have aong (note that this order can be different from λ). A horizonta ine beow is chosen as the spine of Γ and is denoted as. Let δ be the distance between the eftmost vertex and the rightmost vertex of Γ aong spine. Choose the distance between and greater than 3δ. Aso, choose an interva I on of size at most δ. Every vertex v of G has a source position s(v) defined by the point aong representing v in Γ and a target position t(v) on such that t(v) wi represent v in Γ. The target positions are chosen inside interva I in such a way that their eft-to-right order corresponds to λ. Aso, the endpoints of every arc a that Agorithm LinearOrderDraw wi draw either in top or in the bottom page of the ower spine wi be points inside interva I. The trajectory of vertex v is the straight-ine segment s(v)t(v) and it is denoted as τ(v). Agorithm LinearOrderDraw visits the vertices of Γ in the eft-to-right order aong and executes n steps. At each step, a vertex is moved to its target position on and a panar 2-spine drawing with upper spine and ower spine of G is computed. More precisey, a sequence Γ 0,..., Γ n of panar 2-spine drawings with spines and are computed such that Γ 0 coincides with Γ and Γ n coincides with Γ. At Step i (0 i n 1), the panar 2-spine drawing Γ i is transformed into the panar 2-spine drawing Γ i+1 by moving v i to its target position on. When moving vertex v i to its target position, Agorithm LinearOrderDraw maintains the panar embedding of Γ and changes ony the shape of those edges incident on v i and the shape 12

14 of every edge that is intersected by the trajectory of v i. Detais on how the shapes of these edges are changed are given beow. s(v i ) z 0 z 1 y 1 y 0 s(v i ) a 0 a1 a 0 a 2 a 3 a 2 a 3 a 1 y 0 y 1 y 2 y 3 t(v i ) z 3 z 2 y 2 y 3 t(v i ) z 3 z 2 z 1 z 0 (a) (b) s(v i ) z 0 z 1 y 1 y 0 s(v i ) t(v i ) y 0 y 1 y 2 y 3 p 3 p 2 p 1 p 0 q 0 q 1 q 2 q 3 z 3 z 2 y 2 y 3 t(v i ) p 3 p 2 p 1 p 0 q 0 q 1 q 2 q 3 z 3 z 2 z 1 z 0 t t (c) t t (d) Figure 6: Iustration of Step i of Agorithm LinearOrderDraw: Transformation of the shape of the edges intersected by the trajectory τ(v i ) of v i. The trajectory is the ight grey segment. (a) and (c) describe the change of the shapes of eft inter-spine segments and of arcs in the ower sequence. (b) and (d) describe the change of the shapes of right inter-spine segments and of arcs in the ower sequence. Transformation of the shape of the edges intersected by the trajectory of v i. The trajectory τ(v i ) can intersect both inter-spine segments and arcs of the ower sequence of some edges. Let a 0, a 1,..., a h 1 be the sub-edges crossed by τ(v i ) in the order they are encountered when going from s(v i ) to t(v i ) aong τ(v i ). If a j is an arc, denote its endpoints on as y j and z j and assume y j to the eft of z j. If a j is an inter-spine segment and the endpoint of a j that is on is to the eft of t(v i ) denote this endpoint as y j, the other one as z j, and ca the inter-spine segment a eft inter-spine segment (see aso Figure 6(a)); if, otherwise, the endpoint of a j that is on is to the right of t(v i ) denote this endpoint as z j, the other one as y j, and ca the inter-spine segment a right inter-spine segment (see aso Figure 6(b)). Agorithm LinearOrderDraw modifies the shape of the h sub-edges a 0, a 1,..., a h 1 intersected by τ(v i ) as foows. Refer to Figures 6(c) and 6(d). Let t and t be two points of I such that t, t(v i ) and t appear in this eft-to-right order aong and no vertex or spine crossing is between t and t(v i ) and between t(v i ) and t on. Choose h points p 0, p 1,..., p h such that each p j (0 j h) is between t and t(v i ) on and p j is to the right of p j+1 on (0 j h 1). Choose h points q 0, q 1,..., q h such that each q j (0 j h) is between t(v i ) and t on and q j is to the eft of q j+1 on (0 j h 1). If a j is an arc it is repaced by: (i) an arc with endpoints y j and p j ; (ii) an arc with endpoints p j and q j ; (ii) an arc with endpoints q j and z j. If a j is a eft inter-spine segment it is repaced by: (i) an arc with endpoints y j and p j ; (ii) an arc with endpoints p j and q j ; (iii) an inter-spine segment with endpoints q j and z j (Figure 6(c)). If a j is a right inter-spine segment it is repaced by: (i) an inter-spine segment with endpoints y j and p j ; (ii) an arc with endpoints p j and q j ; (iii) an arc with endpoints q j and z j (Figure 6(d)). 13

15 Transformation of the shape of the edges incident on v i. Partition the edges incident on v i in the drawing Γ into four sets. The set E t, (E b, ) contains the edges e = (v j, v i ) such that j < i and the arc of e incident on v i is in the top (bottom) page of the spine of Γ. Anaogousy, E t,r (E b,r ) contains the edges e = (v j, v i ) such that i < j and the arc of e incident on v i is in the top (bottom) page of. Let e = (v j, v i ) be an edge of E t, or E b,. Refer to Figure 7. When v i is moved to its target position, v j has aready been processed and moved to its target position on during a previous step of Agorithm LinearOrderDraw because j < i and the agorithm processes the vertices of Γ in a eft-to-right order. Hence, when going from v j to v i aong e in Γ i we find the (possiby empty) ower sequence σ of e, the inter-spine segment a of e, and the (possiby empty) upper sequence σ u of e. Let x be the endpoint of a on. Repace a and σ u with an arc whose endpoints are x and t(v i ). v i s(v i ) s(v i ) t(v i ) x t(v i ) (a) v x i (b) Figure 7: Iustration of Step i of Agorithm LinearOrderDraw: Transformation of the shape of the edges incident on v i and beonging to E t, or E b,. Let e = (v i, v j ) be an edge of E b,r. Refer to Figure 8). Edge e is represented in Γ i as an arc a with endpoints s(v i ) and s(v j ). Arc a is repaced by the straight-ine segment t(v i )s(v j ). Let e j = (v i, v ij ) (0 j h 1) be the edges of E t,r with i j < i j+1 (0 j < h 1). Refer to Figure 8. Let s be a point on such that s is to the right of s(v i ) and no vertex or spine crossing is between s(v i ) and s on. Choose h points p 0, p 1,..., p h 1 such that each p j (0 j h 1) is between s(v i ) and s on and p j is to the eft of p j+1 aong (0 j < h 1). Edge e j is represented in Γ i as an arc a j with endpoints s(v i ) and s(v ij )(0 j h 1). Arc a j is repaced by the segment t(v i )p j and the arc with endpoints p j and s(v ij ). v i v i1 v j v h v p 0 p 1 v i1 v j v h v v i0 s v i0 s(v i ) s(v i ) t(v i ) t(v i ) v i (a) (b) Figure 8: Iustration of Step i of Agorithm LinearOrderDraw: Transformation of the shape of the edges incident on v i and beonging to E t,r or E b,r. 14

16 6.2 Anaysis of Agorithm LinearOrderDraw In this section we prove the correctness of Agorithm LinearOrderDraw and anayze its time compexity. As expained in the previous section, Agorithm LinearOrderDraw computes first a monotone topoogica book embedding Γ 0 and then it executes n steps to transform Γ 0 into a 3-chain topoogica book embedding. We distinguish each of these n steps with an index i such that 0 i n 1; reca that Step i computes a drawing denoted as Γ i+1. Aso, we sha conventionay denote as Step ( 1) the initia step that computes Γ 0. Let a be a sub-edge of a drawing Γ i that is repaced in Γ i+1 by other sub-edges, and et a be one of these sub-edges. We say that a repaces a; we aso say that a is a repacing sub-edge of Step i. We start by proving that the output of each step is a 2-spine drawing. Lemma 3 Let Γ i be the drawing computed by Step (i 1) of Agorithm LinearOrderDraw (0 i n 1). Γ i is a 2-spine drawing. Proof: Step ( 1) computes a monotone topoogica book embedding Γ 0 by using Theorem 4. By definition, a monotone topoogica book embedding is aso a 2-spine drawing. Assume by induction that the drawing Γ i computed by Step (i 1) (1 i n 1) is a 2-spine drawing. The vertices of Γ i+1 are either points of the ower or of the upper spine by construction. Step i of Agorithm LinearOrderDraw modifies the shape of those edges that are intersected by the trajectory of v i and of those edges that are incident to v i. Let e be an edge of Γ i that is intersected by the trajectory of v i. Agorithm LinearOrderDraw either repaces an arc of e with three arcs or it repaces the inter-spine segment of e with two arcs and a new inter-spine segment (see aso Figure 6); in both cases any two consecutive arcs are on opposite pages. Let e be an edge of Γ i incident on v i. If e E t, or e E b,, then after moving v i to its target position e has both endvertices on a same spine; in this case Agorithm LinearOrderDraw repaces the inter-spine segment of e and the upper sequence of e (if such a sequence exists) with an arc having both endpoints on (see aso Figure 8); if e E t,r or e E b,r, then after moving v i to its target position, edge e has its endvertices on different spines; in this case Agorithm LinearOrderDraw repaces an arc of e having both endpoints on with an inter-spine segment pus (possiby) another arc (see aso Figure 8); in both cases the new shape of e respects the definition of 2-spine drawing. It foows that Γ i+1 is aso a 2-spine drawing. To compete the prof of correctness of Agorithm LinearOrderDraw, we wi first prove that each 2-spine drawing Γ i computed by Step (i 1) is a panar drawing (Lemma 4), and then show that Γ n is a 3-chain monotone topoogica book embedding such that the inear order of the vertices aong the spine respects the given inear order (Lemma 5). The next properties are used to prove the panarity of Γ i. We use the same notation and terminoogy as in the previous section. Property 1 The distance between and and the interva I on are such that: (i) the trajectory of any vertex intersects an arc with endpoints p and q ony if one of the endpoints of the trajectory is in the cosed interva defined by p and q; (ii) no two arcs such that one has its endpoints in the ower spine and the other has its endpoints in the upper spine can intersect. Proof: Let Γ 0 be the monotone topoogica book embedding computed at Step ( 1) of Agorithm LinearOrderDraw. Let δ be the distance between the eftmost vertex and the rightmost vertex of Γ 0 aong spine. The distance between and is chosen to be greater than 3δ and the interva I is chosen to have ength at most δ. Since an arc of an edge with endpoints p and q is drawn as a circuar arc passing trough p, q, and a point of the perpendicuar bisector of pq at a distance d(p,q) 4 from or, a tangent ines to each arc have sope tan π 6 σ tan π 6. By choosing the distance between and greater than 3δ we have that the sope of each trajectory is either ower than tan π 6 or greater than tan π 6. This impies that a trajectory intersects an arc with endpoints p and q ony if one of the endpoints of the trajectory is in the cosed interva defined by p and q. Aso, et a be an arc of any of the drawings computed by any of the steps of Agorithm LinearOrderDraw and et p, q be the endpoints of a. By construction, p and q are inside interva I and therefore we have d(p, q) δ, which impies that the maximum distance between a point of a 15

17 and the spine is at most δ 4. Since the distance between and is arger than δ 2 we have that no two arcs such that one has its endpoint in the ower spine and the other has its endpoints in the upper spine can intersect. Property 2 Let Γ i and Γ i+1 be the 2-spine drawings computed by Steps (i 1) and i of Agorithm LinearOrderDraw, respectivey (0 i n 1). Every arc in the bottom page of the upper spine of Γ i+1 is aso an arc in the bottom page of the upper spine of Γ i. Proof: Step i of Agorithm LinearOrderDraw (i = 0, 1,..., n 1) can change the shape of some edges of Γ i by creating new inter-spine segments and new arcs. These arcs can have endpoints on the ower spine (see aso Figure 6) or can be arcs in the top page of the upper spine (see aso Figures 7 and 8). No arcs in the bottom page of the upper spine are created at Step i. Property 3 Let Γ i be the 2-spine drawing computed by Step (i 1) of Agorithm LinearOrderDraw (0 i n 1) and et τ(v i ) be the trajectory of vertex v i processed at Step i. Every arc of Γ i intersected by τ(v i ) is in the top page of the ower spine of Γ i. Proof: Let a be an arc of Γ i intersected by τ(v i ). Since τ(v i ) is a straight-ine segment with one endpoint in the upper spine and the other endpoint in the ower spine, arc a can either be in the top page of the ower spine or in the bottom page of the upper spine. Assume that a is an arc in the bottom page of the upper spine. By Property 2, a is aso an arc of Γ 0. Since, by Theorem 4, Γ 0 is a monotone topoogica book embedding, if a is in the bottom page of the upper spine then the eftmost endpoint of a is a vertex of the input graph G, that we denote as v j. Agorithm LinearOrderDraw defines the distance between the two spines and and the target positions aong in such a way that for every vertex v with source position s(v) and target position t(v), the trajectory τ(v) intersects a ony if s(v) is in the interva between the endpoints of a. It foows that vertex v j is eft of v i aong the spine of Γ 0, that is j < i. Since Agorithm LinearOrderDraw processes the vertices in the eft-to-right order aong the spine of Γ 0, vertex v j is moved to its target position before Step i is executed. Aso, when the eftmost endvertex of an arc beonging to the to the bottom page of the upper spine is moved to its target position, then this arc is repaced by an inter-spine segment (see aso Figure 8). It foows that a cannot be an arc of Γ i such that a is in the bottom page of the upper spine and a is intersected by τ(v i ). Property 4 Let Γ i+1 be the 2-spine drawing computed by Step i of Agorithm LinearOrderDraw (0 i n 1). Let a be an arc of Γ i+1 in the top page of the ower spine. Let y and z be the endpoints of a, with y to the eft of z. Point t(v i ) cannot be a point to the right of y and to the eft of z. Proof: Two cases are possibe: Either a is an arc in the top page of the ower spine aso in the drawing Γ i computed by Step (i 1) or a is created at Step i. In the first case, a cannot be crossed by τ(v i ) (because otherwise a woud not exist in Γ i+1 ) and thus the property immediatey hods. In the second case, either t(v i ) is an endpoint of a or a is a repacing sub-edge of Step i. Then, by construction the endpoints of a are either both to the eft of t(v i ), or both to the right of t(v i ) (see aso Figure 6). Property 5 Let Γ i+1 be the 2-spine drawing computed by Step i of Agorithm LinearOrderDraw (0 i n 1). Let a be an arc of Γ i+1 in the bottom page of the ower spine. Let y and z be the endpoints of a, with y to the eft of z. If a is a repacing sub-edge of Step i, point t(v i ) is to the right of y and to the eft of z. Proof: Since a is a repacing sub-edge of Step i, then it repaces a sub-edge a that is crossed by τ(v i ). Both in the case when a is an arc and in the case when a is an inter-spine segment, the ony sub-edge that repaces a and is in the bottom page has t(v i ) between its endpoint (see aso Figure 6). 16