Proximity constraints and representable trees (extended abstract)

Bose, Prosenjit; Battista, Giuseppe Di; Lenhart, William; Liotta, Giuseppe

doi:10.1007/3-540-58950-3_389

Cited by 22 publications

(23 citation statements)

References 22 publications

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“…Consider the graph G I that has an s-3-drawing with the same set of points for the vertices as F. From Property 2 we have that G C G I. From [3] we have that in the above interval of 3 values, s-3-drawable graphs are planar; hence, G I is planar. The conclusion follows from the maximality of G.…”

Section: Sketch Of Proofmentioning

confidence: 99%

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The strength of weak proximity (extended abstract)

Battista¹,

Liotta

Whitesides

1996

Lecture Notes in Computer Science

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Abstract. 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.

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Section: Sketch Of Proofmentioning

confidence: 99%

“…By relaxing (ii), a graph G can no longer be reconstructed from the locations of its vertices in a weak drawing; however, many graphs that do not admit strong drawings can be drawn weakly. For example, a tree that has a vertex of degree greater than five has no strong E-drawing for any ~ [3]. Thus the drawing in Fig.…”

Section: Introduction and Overviewmentioning

confidence: 99%

The strength of weak proximity (extended abstract)

Battista¹,

Liotta

Whitesides

1996

Lecture Notes in Computer Science

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“…Since a graph may, for a given value of t3, have many--very different--13-drawings, it is also of interest to examine the "aesthetic" properties of such drawings. In [3,2], several results concerning 13-drawings of trees are presented.…”

Section: A Delaunay Drawingmentioning

confidence: 99%

“…Open and closed strip drawings of trees were investigated in [2]. Closed strip drawable trees were completely characterized and a linear time drawing algorithm was given.…”

Section: /~-Drawingsmentioning

confidence: 99%

Proximity drawability: A survey extended abstract

1995

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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

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“…In [2] a complete characterization is given of those trees that admit a proximity drawing, for each of the following types of proximity region: (i) relative neighborhood region, (ii) relatively closest region (iii) modified Gabriel region, and (iv) Gabriel region. In [3], the authors studied the proximity drawability testing problem for trees for an infinite family of parametrized proximity regions, called fl-regions, that include open and closed disks and lunes as special cases. 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].…”

Section: Spectively) Obtained As the Intersection Of The Two Disks Wmentioning

confidence: 99%

Recognizing rectangle of influence drawable graphs (extended abstract)

et al. 1995

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Abstract. Given two points x and y in the plane the rectangle of influence of x and y is the axis-aligned rectangle having x and y at opposite corners. The rectangle of influence drawing of a graph G is a straight-line drawing of G such that (i) for each pair of adjacent vertices u, v of G the rectangle of influence of the points representing u and v is empty (i.e. does not contain any other point representing any other vertex of G except possibly the two representing u and v) and (ii) for each pair of non adjacent vertices u, v of G the rectangle of influence of the points representing u and v is not empty. In this paper we consider several classes of graphs, and we characterize, for each class, which graphs of the class have a rectangle of influence drawing. For each class we show that the problem of testing whether a graph G has a rectangle of influence drawing can be done in linear time. Furthermore, if the test for G is affirmative, a rectangle of influence drawing of G can be constructed in linear time with the real RAM model of computation.

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Proximity constraints and representable trees (extended abstract)

Cited by 22 publications

References 22 publications

The strength of weak proximity (extended abstract)

The strength of weak proximity (extended abstract)

Proximity drawability: A survey extended abstract

Recognizing rectangle of influence drawable graphs (extended abstract)

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