We introduce a novel technique for drawing proximity graphs in polynomial area and volume. Previously known algorithms produce representations whose size increases exponentially with the size of the graph. This holds even when we restrict ourselves to binary trees. Our method is quite general and yields the first algorithms to construct polynomial area weak Gabriel drawings of ternary trees, polynomial area weak $\beta$-proximity drawing of binary trees for any $0 \beta < \infty$, and polynomial volume weak Gabriel drawings of unbounded degree trees. Notice that, in general, the above graphs do not admit a strong proximity drawing. Finally, we give evidence of the effectiveness of our technique by showing that a class of graph requiring exponential area even for weak Gabriel drawings, admits a linear volume strong $\beta$-proximity drawing and a relative neighborhood drawing. All the algorithms described run in linear time.

We show that any tree whose maximum degree is at most 10 can be drawn in 3-space such that it is the minimum spanning tree of its vertices. 1

. In this paper we investigate the area requirement of proximity drawings and we prove an exponential lower bound. Our main contribution is to show the existence of a class of Gabriel-drawable graphs that require exponential area for any Gabriel drawing and any resolution rule. The result is further extended to an infinite class of proximity drawings. 1 Introduction. Proximity drawings of graphs have received increasing attention recently in the computational geometry and graph drawing communities due to the large number of applications where they play a crucial role. Such applications include pattern recognition and classification, geographic variation analysis, geographic information systems, computational geometry, computational morphology, and computer vision (see, e.g. [23, 21, 26]). A proximity drawing is a straight--line drawing where two vertices are adjacent if and only if they are neighbors according to some definition of neighborhood. ? Work supported in part by the US Na...

A graph is minimum weight drawable if it admits a straight-line drawing that is a minimum weight triangulation of the set of points representing the vertices of the graph. In this paper we consider the problem of characterizing those graphs that are minimum weight drawable. Our contribution is twofold: We show that there exist infinitely many triangulations that are not minimum weight drawable. Furthermore, we present non-trivial classes of triangulations that are minimum weight drawable, along with corresponding linear time (real RAM) algorithms that take as input any graph from one of these classes and produce as output such a drawing. One consequence of our work is the construction of triangulations that are minimum weight drawable but none of which is Delaunay drawable---that is, drawable as a Delaunay triangulation. 1 Introduction and Overview Recently much attention has been devoted to the study of combinatorial properties of well-known geometric structures---often referred to a...

In this paper, we study weak &beta;-proximity drawings. All known algorithms that compute (weak) proximity drawings produce representations whose area increases exponentially with the number of vertices. Additionally, an exponential lower bound on the area of (weak) proximity drawings of general graph has been proved. We present the first algorithms that compute a polynomial area &beta;-proximity drawing of binary and ternary trees. The algorithms run in linear time.

A family of proximity drawings of graphs called open and closed &beta;-drawings, first defined in [16], and including the Gabriel, relative neighborhood and strip drawings, are investigated. Complete characterizations of which trees admit open &beta;-drawings for 0 &le &beta; &le; ... or closed &beta;-drawings for 0 &le; &beta; ... are given, as well as partial characterizations for other values of &beta;. For &beta; < &infin; in the intervals in which complete characterizations are given, it can be determined in linear time whether a tree admits an open or closed &beta;-drawing, and, if so, such a drawing can be computed in linear time in the real RAM model. Finally, a complete characterization of all graphs which admit closed strip drawings is given.

This paper initiates the study of weak proximity drawings of graphs and demonstrates their advantages over strong proximity drawings in certain cases. Weak proximity drawings are straight line drawings such that if the proximity region of two points p and q representing vertices is devoid of other points representing vertices, then segment (p, q) is allowed, but not forced, to appear in the drawing. This differs from the usual, strong, notion of proximity drawing in which such segments must appear in the drawing. Most previously studied...

In this paper we describe a general technique for establishing NP-hardness of graph representations. This technique is a generalization of the tool called the logic engine. We show that it is possible to extend it to a wobbly logic engine, which provides a proof method of NP-hardness for a variety of graph representations for which the set of feasible representations does not have to be discrete. This includes representations by visibility and intersection. In particular, we give a first proof that it is NP-hard to decide whether a graph has a nondegenerate z-axis parallel visibility representation (ZPR) by unit squares.

Visualization techniques for complex data are a workhorse of modern scientific pursuits. The goal of visualization is to embed high-dimensional data in a low-dimensional space while preserving structure in the data relevant to exploratory data analysis such as clusters. However, existing visualization methods often either fail to separate clusters due to the crowding problem or can only separate clusters at a single resolution. Here, we develop a new approach to visualization, tree preserving embedding (TPE). Our approach uses the topological notion of connectedness to separate clusters at all resolutions. We provide a formal guarantee of cluster separation for our approach that holds for finite samples. Our approach requires no parameters and can handle general types of data, making it easy to use in practice. 1.