Directed Rectangle-Visibility Graphs have. Abstract. Visibility representations of graphs map vertices to sets in Euclidean space and

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1 Direted Retangle-Visibility Graphs have Unbounded Dimension Kathleen Romanik DIMACS Center for Disrete Mathematis and Theoretial Computer Siene Rutgers, The State University of New Jersey P.O. Box 1179, Pisataway, NJ Abstrat Visibility representations of graphs map verties to sets in Eulidean spae and express edges as visibility relations between these sets. One visibility representation in the plane that has been studied is one in whih the verties of the graph map to losed isotheti retangles and the edges are expressed by horizontal or vertial visibility between the retangles. Two retangles are only onsidered to be visible to one another if there is a non-zero width horizontal or vertial band of sight between them. A graph that an be represented in this way is alled a retangle-visibility graph. A retangle-visibility graph an be direted by direting all edges towards the positive x and y diretions, whih yields a direted ayli graph. A direted ayli graph G has dimension d if d is the minimum integer suh that the verties of G an be ordered by d linear orderings, < 1 ; : : :; < d, and for verties u and v there is a direted path from u to v if and only if u < i v for all 1 i d. In this note we show that the dimension of the lass of direted retangle-visibility graphs is unbounded. 1 Retangle-Visibility Graphs The problem of determining a visibility representation of a graph, where the verties of the graph map to sets in Eulidean spae and the edges are expressed as visibility relations between these sets, has been widely studied (see [BETT93] for a survey). One representation in the plane that has been studied [Wis89, DH94, DH96] maps eah vertex of the graph to a losed isotheti retangle in E 2 and eah edge to a horizontal or vertial band of sight between two retangles. DIMACS is an NSF Siene and Tehnology Center, funded under ontrat STC , and also reeives support from the New Jersey Commission on Siene and Tehnology. 1

2 More formally, onsider an arrangement of losed retangles in E 2 suh that the sides of the retangles are parallel to the axes and the retangles are pairwise interior disjoint. Two retangles R i and R j are -visible if there is a non-degenerate retangle R ij with two opposite sides that are subsets of the boundaries of R i and R j, and R ij intersets no other retangle. Suh an arrangement is a retangle-visibility representation of a graph G = (V; E). A graph admits suh a representation provided that the following hold: There exists a 1-1 onto orrespondene between the retangles and the verties in V. Verties v i and v j are adjaent in G if and only if their orresponding retangles R i and R j are -visible. Retangle-visibility graphs are an extension of bar-visibility graphs, whih were dened independently by Wismath [Wis85] and Tamassia and Tollis [TT86]. In this representation verties map to losed, disjoint, horizontal line segments in the plane, and two verties are adjaent in the graph if and only if their orresponding segments are -visible in the vertial diretion. The lass of all bar-visibility graphs was ompletely haraterized by both Wismath [Wis85] and Tamassia and Tollis [TT86]. They independently proved that a graph has a bar-visibility representation if and only if it has a planar embedding suh that all ut verties lie on the external fae. The lass of all retangle-visibility graphs has not been ompletely haraterized. However, Wismath [Wis89] proved that all planar graphs have a retangle-visibility representation. Also, Dean and Huthinson [DH94, DH96] proved that a omplete bipartite graph K p;q has a retangle-visibility representation if and only if p 4. Another extension of bar-visibility graphs that was studied by Bose et al. [BEF + 94] is the lass of VR-representable graphs. In this representation eah vertex of the graph maps to a losed retangle in E 3 and edges are expressed by vertial visibility between retangles. The retangles representing verties are disjoint, ontained in planes perpendiular to the z-axis, and have sides parallel to the x or y axes. Two verties are adjaent in the graph 2

3 if and only if their orresponding retangles are -visible in the z diretion. 2 Dimension of Direted Ayli Graphs A direted ayli graph G has dimension d if d is the minimum integer suh that the verties of G an be ordered by d linear orderings, < 1 ; : : :; < d, and for verties u and v there is a direted path from u to v if and only if u < i v for all 1 i d [Tro92]. A lass G of graphs has dimension d if d is the largest dimension of any graph in G. A bar-visibility representation of a graph an be direted by direting all edges towards the positive y diretion. It has been shown ([BT88, RU88]) that any graph with a direted bar-visibility representation has dimension at most two. Both extensions of bar-visibility graphs dened in the previous setion an be direted, yielding direted ayli graphs. A VR-representation of a graph is direted by direting all edges towards the positive z diretion. A retangle-visibility representation of a graph is direted by direting all edges towards the positive x and y diretions. Romanik [Rom94] proved that the dimension of the lass of direted VR-representable graphs is unbounded. In this note we prove that the dimension of the lass of direted retangle-visibility graphs is unbounded. Sine a direted ayli graph an be used to represent a partial order, work done by Rival and Urrutia [RU88, RU92] on representing partially ordered sets by moving onvex objets in spae is related to our study of the dimension of retangle-visibility graphs. 3 Unbounded Dimension of Retangle-Visibility Graphs We show that the dimension of the lass of direted retangle-visibility graphs is unbounded by giving a lass of graphs G = fg n j n 1g suh that the dimension of G n is at least n, and then giving a direted retangle-visibility representation of G n. Let us denote by K n;n?m a omplete bipartite graph with a perfet mathing removed, 3

4 where n is the size of both partitions. Note that both partitions must have the same size for there to be a perfet mathing. It is well known that the direted K n;n? M, where all edges are direted from one partition to the other, has dimension n. The direted graph G n = (V; E) that we onstrut is similar to K n;n? M, exept that the edges are replaed by direted paths. It has 2n + 3n(n? 1) verties dened as follows: V = fa 1 ; : : :; a n ; e 1 ; : : :; e n g [ fb i;j ; i;j ; d i;j j 1 i; j n; i 6= jg The verties a 1 ; : : :; a n and e 1 ; : : :; e n orrespond to the two partitions of K n;n? M. The following is a desription of the edges of G n : Eah vertex a i is a soure and has edges f(a i ; b i;j ) j j 6= ig oming out of it. The b i;j ; i;j ; d i;j verties are onneted by edges (b i;j ; i;j ) and ( i;j ; d i;j ). Eah vertex e j is a sink and has edges f(d i;j ; e j ) j j 6= ig going into it. See Figure 1 for an illustration of the subgraph of G n with soure vertex a i. In graph G n eah vertex a i has a direted path to eah e j, where j 6= i, but there is no path from a i to e i. Lemma 3.1 Graph G n has dimension at least n. Proof: We onsider only the relative order of the a i and e i verties. Sine there is a direted path from a i to e j ; j 6= i, a i must appear before e j ; j 6= i (i.e. a i < e j ) in eah linear ordering of the verties. Sine there is no path from a i to e i, e i must appear before a i (i.e. e i < a i ) in some linear ordering of the verties. Consider a linear ordering < l in whih e i < l a i. For all other a j ; j 6= i, we must have a j < l e i, and for all other e j ; j 6= i, we must have a i < l e j. Thus in the ordering < l, no other pair a j ; e j an be reversed. Sine eah pair a j ; e j must be reversed in some ordering, this requires at least n linear orderings. ut 4

5 e... 1 e i-1 e... i e i+1 e n d i,1 d i,i-1 d i,i+1 d i,n i,1 i,i-1 i,i+1 i,n b i,1 b i,i-1 b i,i+1 b i,n a i Figure 1: Subgraph of G n with soure vertex a i We now desribe a direted retangle-visibility representation for G n. The retangles for the a i verties are in a stairstep arrangement, as are the retangles for the b i;j verties, the d i;j verties and the e i verties (see Figure 2). The vertial edges of the b i;j retangles and the horizontal edges of the d i;j retangles, if extended, interset to form a grid pattern. The i;j retangles are formed by \lling in" squares of this grid. It is easy to see that any b i;j vertex an be onneted to any d l;k vertex by lling in the appropriate square of the grid. All of the (a i ; b i;j ) edges and ( i;j ; d i;j ) edges are horizontal, and the rest of the edges are vertial. Figure 2 illustrates the onstrution for G 4. It is easy to see how this onstrution an be extended for any n 1. Using this retangle-visibility representation of G n we get the following theorem. Theorem 3.1 The dimension of the lass of direted retangle-visibility graphs is unbounded. 5

6 e 1 e 2 e 3 e 4 a 1 a 2 a 3 a 4 Referenes Figure 2: Retangle-visibility representation for graph G n [BEF + 94] P. Bose, H. Everett, S. Fekete, A. Lubiw, H. Meijer, K. Romanik, T. Shermer, and S. Whitesides, On a visibility representation for graphs in three dimensions, in: D. Avis and P. Bose, eds., Snapshots in Computational and Disrete Geometry, Volume III, MGill University Tehnial Report SOCS (1994). [BETT93] G. Di Battista, P. Eades, R. Tamassia, and I.G. Tollis, Algorithms for automati graph drawing: An annotated bibliography, Tehnial report, Department of Computer Siene, Brown University (1993). [BT88] [DH94] G. Di Battista and R. Tamassia, Algorithms for plane representations of ayli digraphs, Theoretial Computer Siene 61 (1988) 175{198. A.M. Dean and J.P. Huthinson. Retangle-visibility representations of bipartite graphs, in: R. Tamassia and I. Tollis, eds., Graph Drawing '94, Prineton, 6

7 NJ (1994) 159{166. [DH96] [RU88] [RU92] [Rom94] [Tro92] [TT86] A.M. Dean and J.P. Huthinson. Retangle-visibility representations of bipartite graphs, Disrete Applied Mathematis { to appear. I. Rival and J. Urrutia, Representing orders by translating onvex gures in the plane, Order 4 (1988) 319{339. I. Rival and J. Urrutia, Representing orders by moving gures in spae, Disrete Mathematis 109 (1992) 255{263. K. Romanik, Direted VR-representable graphs have unbounded dimension, in: R. Tamassia and I. Tollis, eds., Graph Drawing '94, Prineton, NJ (1994) 177{181. W.T. Trotter, Combinatoris and Partially Ordered Sets: Dimension Theory (Johns Hopkins University Press, Baltimore, MD, 1992). R. Tamassia and I.G. Tollis, A unied approah to visibility representations of planar graphs, Disrete Computational Geometry 1 (1986) 321{341. [Wis85] S.K. Wismath, Charaterizing bar line-of-sight graphs, in: Proeedings of the First Annual Symposium on Computational Geometry, (ACM Press, 1985) 147{152. [Wis89] S.K. Wismath. Bar-representable visibility graphs and a related ow problem, Ph.D. Thesis, University of British Columbia, Vanouver, BC (1989). 7