J. P. Hutchinson, T. Shermer, and A. Vince, On representations of some thickness-two graphs. Comput. Geom., 13(3):161-171, 1999.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....to prove the main result in Section 3. 1.1 Related Work Since every planar graph has a drawing in the plane with straight line edges [13, 24] the graphs with geometric thickness 1 are precisely the planar graphs. Graphs with geometric thickness 2, called doubly linear graphs, are studied in [5, 16]. Hutchinson et al. 16] prove that every doubly linear graph has at most 6n 18 edges, and present doubly linear graphs with 6n 20 edges for all n 8. Dillencourt et al. 11] establish lower and upper bounds for the geometric thickness of complete and complete bipartite graphs. It is shown that ....

....in Section 3. 1.1 Related Work Since every planar graph has a drawing in the plane with straight line edges [13, 24] the graphs with geometric thickness 1 are precisely the planar graphs. Graphs with geometric thickness 2, called doubly linear graphs, are studied in [5, 16] Hutchinson et al. [16] prove that every doubly linear graph has at most 6n 18 edges, and present doubly linear graphs with 6n 20 edges for all n 8. Dillencourt et al. 11] establish lower and upper bounds for the geometric thickness of complete and complete bipartite graphs. It is shown that d n 1 5:646 e (Kn ) ....

J. P. Hutchinson, T. Shermer, and A. Vince, On representations of some thickness-two graphs. Comput. Geom., 13(3):161-171, 1999.

....a straight line drawing and two subgraphs G 1 and G 2 whose union is G, and if the straight line drawing of G restricted to G i is a planar embedding of G i , for 1 # i # 2, then G is called doubly linear . Clearly any doubly linear graph has thickness at most two. Hutchinson et al. HSV96, HSV99] study doubly linear graphs. They show that a doubly linear graph with n vertices has at most 6n 18 edges. Other variations of thickness are discussed in [Hob69, H S82, Hor83, Wes83a, PCK89, DEH00] for example. 6 Crossing Number In graph drawing, but also in other application areas such ....

Joan P. Hutchinson, Thomas Shermer, and Andrew Vince. On representations of some thickness-two graphs. Computational Geometry, 13:161--171, 1999.

....G has a straight line drawing and two subgraphs G 1 and G 2 whose union is G, and if the straight line drawing of G restricted to G i is a planar embedding of G i , for 1 # i # 2, then G is called doubly linear . Clearly any doubly linear graph has thickness at most two. Hutchinson et al. HSV96, HSV99] study doubly linear graphs. They show that a doubly linear graph with n vertices has at most 6n 18 edges. Other variations of thickness are discussed in [Hob69, H S82, Hor83, Wes83a, PCK89, DEH00] for example. 6 Crossing Number In graph drawing, but also in other application areas ....

Joan P. Hutchinson, Thomas Shermer, and Andrew Vince. On Representations of Some Thickness-Two Graphs. In Franz J. Brandenburg, editor, Graph Drawing. Proceedings of the DIMACS International Workshop, GD'95, pages 324--332. Springer-Verlag, Lecture Notes in Computer Science, vol. 1027, 1996.

....have L(u) L(v) Otherwise, the last used re evaluation rule on v is 1 b) or 2 b) and L(u) L(v) 5 2 Visibility representation For this type of drawing only vertical and horizontal lines are allowed for the edges and vertices are rectangles. This type of representation has been proposed in [22, 20, 7, 6, 21]. A grid of size (n 1) n 1) is used in [2] to obtain such kind of drawings with a linear time algorithm for planar graphs. Here, we present new linear time algorithm for 2 visibility drawings. This algorithm is based on the construction of a strati cation L T of a planar graph G. Such kind ....

T. Sher;er J.P. Hutchinson and A. Vince. On representation of some thickness-two graphs. In S. Whitesides, editor, Graph Drawing (Proc. GD '95), volume 1027 of Lecture Notes in Computer Science, pages 324{ 332. Spinger-Verlag, 1996.

....of G as a line segment, and assign each edge to one of k layers so that no two edges on the same layer cross. This corresponds to the notion of real linear thickness introduced by Kainen [15] Graphs with geometric thickness 2 (called doubly linear graphs) have been studied by Hutchinson et al. [13], where the connection with certain types of visibility graphs was explored. A notion related to geometrical thickness is that of (graph theoretical) thickness of a graph, #(G) which has been studied extensively [1, 3, 8, 9, 10, 14, 16] and has been defined as the minimum number of planar graphs ....

J. P. Hutchinson, T. Shermer, and A. Vince. On representation of some thickness-two graphs. In F. J. Brandenburg, editor, Symposium on Graph Drawing (GD '95), pages 324--332, Passau, Germany, September 1995. Springer-Verlag Lecture Notes in Computer Science 1027.

....As a special case they obtained the following result. Theorem 3.7 [JMOS95] If G is a graph without K 5 minors, then (G) 2. Moreover, graphs with thickness two have drawn some attention in the field of graph drawing, where there are used in the study of so called rectangle visibility graphs [HSV95]. To our knowledge, the thickness of no other graph class has been settled yet. Moreover, we cannot expect to find a nice formula describing the thickness of an arbitrary graph, since the thickness problem was proven to be NP hard by Mansfield [Man83] Hence, we turn to upper bounds. A simple ....

Hutchinson, J., T. Shermer, and A. Vince, On Representations of some Thickness-Two Graphs, Extended Abstract, Lecture Notes in Computer Science, Graph Drawing '95, F. Brandenburg ed., Springer (to appear).

....generalization of the 1 visibility model. Here vertical and horizontal edges are allowed. Figure 2 shows a 2 visibility drawing for the graph of Figure 1. Though the resulting drawings look very promising from a practical point of view, this model has not been considered very often [6, 9, 17, 18] and the known results are very preliminary. Notice that also 2 visibility representations of nonplanar graphs might be possible, though with crossing edges. In this paper we only consider planar graphs and planar representations. The purpose is to introduce this model as a practical alternative ....

Hutchinson, J.P., T. Shermer and A. Vince, On representation of some thickness-two graphs, Proc. 4th Symposium on Graph Drawing (GD'95), LNCS 1027, Springer-Verlag, pp. 324-332, 1996.

....grow arbitrarily (similar as in 2 visibility representations) we can even prove that the grid has width and height m 2 Gamma 3 4 n 3 with m Gamma 6n 20 bends. This is optimal in the number of bends, since any orthogonal drawing has at most 6n Gamma 20 edges drawn as straight lines [9]. Now we prove a lower bound on the number of bends. Theorem 7. Any drawing of the Kn in the Kandinsky model has at least m Gamma n bends. Proof. Assume we have a drawing Gamma of Kn . Let A be the number of vertical slots that contain nodes. Let B be the number of horizontal slots that ....

....to all nodes and the graph stays planar. If we choose this dummy nodes as last node, then we can show that we get an (n Gamma 1) Theta n drawing where all edges are routed horizontally. So the produced drawing is a 1 visibility representation. As opposed to all previous algorithms (e.g. [16,14,9,6]) in our drawings there are known bounds on the height of a node. Theorem 11. Let G be an outer planar graph. Then G has a 1 visibility representation in an (n Gamma 1) Theta n grid. Every node v has height at most deg(v) In our drawings of planar graphs, the half perimeter of each node is at ....

J. Hutchinson, T. Shermer, and A. Vince. On representation of some thickness-two graphs. In F. Brandenburg, editor, Symp. on Graph Drawing, GD 95, volume 1027 of Lecture Notes in Computer Science, pp. 324--332. Springer Verlag, 1996.

....on the required area. 2 visibility representations are a straightforward generalization of the 1 visibility model. Here vertical and horizontal edges are allowed. Though the resulting drawings look very promising from a practical point of view, this model has not been considered very often e.g. [2, 4] and the known results are very preliminary. Notice that also 2visibility representations of nonplanar graphs might be possible, though with crossing edges. Nevertheless we will only consider planar graphs in this paper. The purpose of this paper is to introduce this model as a practical ....

Hutchinson, J.P., T. Shermer and A. Vince, On representation of some thickness-two graphs, Proc. 4th Symposium on Graph Drawing (GD'95), LNCS 1027, Springer-Verlag, pp. 324-332, 1996.

....Technische Universitat Graz, Austria. 1 visibility representation and study its properties. This paper combines experimental results with the results of two conference papers [4, 14] and a technical report [5] which have motivated a number of other conference papers such as [2, 7, 15, 17, 21]. Closely related questions have been examined in [10, 19] This paper introduces the basic concepts and fundamental results in this area. 1.1 Previous results We begin by reviewing some of the results in 2 dimensional visibility representations. To do this, we first discuss in more detail the ....

J. Hutchinson, T. Shermer, and A. Vince. On representations of some thickness-two graphs. Proc. Graph Drawing 95, Passau 1996. Lecture Notes in Computer Science Vol. 1072, Springer-Verlag 1996, pp. 324--332. 15

....Commission on Science and Technology. RVGs. For a more thorough treatment, as well as motivational details, the reader is referred to the full paper. Wismath [6] seems to be the first to have studied RVGs, and showed that every planar graph is a collinear RVG. Hutchinson, Shermer, and Vince [3] established that no RVG has more than 6n Gamma 20 edges (where n is the number of vertices) Dean and Hutchinson [1] determined which complete bipartite graphs are collinear RVGs (K p;q for p 4) and noncollinear RVGs (K p;q for p 2 or p; q = 3; 3) or p; q = 3; 4) Recently, Shermer [4] has ....

J. P. Hutchinson, T. Shermer, and A. Vince. On representations of some thickness-two graphs (extended abstract). Proc. of Graph Drawing 95, LNCS vol. 1027. Springer-Verlag, 1995.

....a representation an RVG layout of the graph. Shermer [11] has shown that it is NP complete to determine if a graph is an RVG, and so it is of interest to determine classes of graphs that are and are not RVGs. There is now a considerable body of research on RVGs; for results and applications see [5, 6, 3, 4, 2] and others. The focus of this paper is those graphs whose edges can be partitioned into particular types of trees (or forests) we say then that the graph is the union of these trees. An RVG is seen to be the union of two planar graphs (i.e. has thickness at most two) by considering the vertical ....

....a union of two BVGs . This leads to the question of when a union of two BVGs is an RVG; our question is when the union of two trees is an RVG. The union of two trees has at most 2n 2 edges, when the union has n vertices, and this edge bound is well below the bound of 6n 20 edges for general RVGs [5, 6]; however, in the same papers it is shown that for each n 9 and m 35 there is a thickness 2 graph with n vertices and m edges that is not an RVG. The union of two trees has clique number at most 4, and it is possible to lay out K 8 as an RVG; however, again it is not hard to construct graphs ....

[Article contains additional citation context not shown here]

J. Hutchinson, T. Shermer, and A. Vince, On representations of some thickness-two graphs (extended abstract), Lecture Notes in Computer Science #1027, (F. Brandenburg, ed.), Springer-Verlag, Berlin, 324-332, 1995.

....a representation an RVG layout of the graph. Shermer [11] has shown that it is NP complete to determine if a graph is an RVG, and so it is of interest to determine classes of graphs that are and are not RVGs. There is now a considerable body of research on RVGs; for results and applications see [5, 6, 3, 4, 2] and others. The focus of this paper is those graphs whose edges can be partitioned into particular types of trees (or forests) we say then that the graph is the union of these trees. An RVG is seen to be the union of two planar graphs (i.e. has thickness at most two) by considering the vertical ....

....a union of two BVGs . This leads to the question of when a union of two BVGs is an RVG; our question is when the union of two trees is an RVG. The union of two trees has at most 2n 2 edges, when the union has n vertices, and this edge bound is well below the bound of 6n 20 edges for general RVGs [5, 6]; however, in the same papers it is shown that for each n 9 and m 35 there is a thickness 2 graph with n vertices and m edges that is not an RVG. The union of two trees has clique number at most 4, and it is possible to lay out K 8 as an RVG; however, again it is not hard to construct graphs ....

[Article contains additional citation context not shown here]

J. Hutchinson, T. Shermer, and A. Vince, On representations of some thickness-two graphs, Computational Geometry: Theory and Applications, to appear.

....in 3 dimensions have been relatively unstudied. In this paper, we define a 3 dimensional visibility representation and study its properties. Earlier conference and technical report versions of this paper ( 2] 3] have motivated a large number of conference papers see [1] 9] 10] [11], 14] 15] Closely related questions have been examined in [5] 13] It is one of our objectives to present a formal version of this early paper of the area. We begin by reviewing some of the results in 2 dimensional visibility representations. To do this, we first discuss in more detail the ....

J. Hutchinson, T. Shermer, and A. Vince. On representations of some thickness-two graphs. To appear in Proc. Graph Drawing 95, Passau 1996. Lecture Notes in Computer Science, Springer-Verlag, 1996.

....less attention. In this paper, we de ne a 3 dimensional visibility representation and study its properties. This paper combines experimental results with the results of two conference papers [4, 14] and a technical report [5] which have motivated a number of other conference papers such as [2, 7, 15, 17, 21]. Closely related questions have been examined in [10, 19] This paper introduces the basic concepts and fundamental results in this area. 1.1 Previous results We begin by reviewing some of the results in 2 dimensional visibility representations. To do this, we rst discuss in more detail the ....

J. Hutchinson, T. Shermer, and A. Vince. On representations of some thickness-two graphs. Proc. Graph Drawing 95, Passau 1996. Lecture Notes in Computer Science Vol.1072, Springer-Verlag 1996, pp. 324-332.

No context found.

J. Hutchinson, T. Shermer, and A. Vince, "On representations of some thickness-two graphs", Proc. Graph Drawing 95, Passau 1996, Lecture Notes in Computer Science #1072, Springer-Verlag 1996, pp. 324-332.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC