Proximity drawability: A survey extended abstract

Abstract

Abstract. Increasing attention has been given recently to drawings of graphs in which edges connect vertices based on some notion of proximity. Among such drawings are Gabriel, relative neighborhood, Delaunay, sphere of influence, and minimum spanning drawings. This paper attempts to survey the work that has been done to date on proximity drawings, along with some of the problems which remain open in this area. Proximity DrawingsIn 1969, Gabriel and Sokal [15] presented a method for associating a graph to a set of geographic data points P by connecting points z, y E P with an edge if and only if the closed disk having the segment ~ as diameter contained no other point of P. This graph, now called the Gabriel graph of P, is just one example of what have come to be called proximity graphs. Loosely speaking, a proximity graph is a graph constructed from a set P of points in some metric space by connecting pairs of points which are deemed to be "sufficiently" close together. A set P can give rise to a variety of different proximity graphs depending upon the definition of closeness used. Early work in this area was concerned for the most part with the problems of determining notions of proximity which might best capture the "internal structure" of a set of points and, having done so, of efficiently computing the proximity graph of a given set of points. For a survey of such results, see Jaromczyk and Toussaint [21].More recently, increasing attention has been given to the proximity drawing problem: given a graph G and a definition of proximity, determine whether a set P of points exists such that the proximity graph of P is the given graph, and if so, compute such a set. Clearly the set P, if it exists, gives rise to a straight-line

Optimal Polygonal Representation of Planar Graphs

“…This is the most extensively studied class of contact graphs, due in part to its relation to application areas such as VLSI floor-planning [25,34], architectural design [30] and geographic information systems [11], but also due to the mathematical ramifications and connections to other areas such as rectangle-ofinfluence drawings [28] and proximity drawings [2,19].…”

Recognizing rectangle of influence drawable graphs (extended abstract)

“…Results that are closely related to the proximity drawability testing problem concern the drawability of trees as minimum spanning trees [10], of triangulations as delaunay triangulations [9], [8], and of planar graphs as nearest neighbor graphs [21]. A survey on the proximity drawability testing problem is in [7].…”

Voronoi Drawings of Trees

“…The problem of deciding whether a graph can be drawn as rectangle of influence graph and the design of efficient algorithms for computing such a drawing when one exists is investigated in [9]. A survey on the problem of drawing a graph as a given type of geometric graph can be found in [3].…”

“…As a contrast, Euclidean Voronoi drawings only require a screen-area that is polynomial in the input size. For references on proximity drawings of graphs see [3].…”