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