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The Rectangle of Influence Drawability Problem
- Computational Geometry: Theory and Applications
, 1997
"... Motivated by rectangular visibility and graph drawing applications, we study the problem of characterizing classes of graphs that admit rectangle of influence drawings. We consider several classes of graphs and show, for each class, that testing whether a graph G has a rectangle of influence draw ..."
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Cited by 12 (3 self)
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Motivated by rectangular visibility and graph drawing applications, we study the problem of characterizing classes of graphs that admit rectangle of influence drawings. We consider several classes of graphs and show, for each class, that testing whether a graph G has a rectangle of influence drawing can be done in O(n) time, where n is the number of vertices of G. If the test for G is affirmative, we show how to construct a rectangle of influence drawing of G. All the drawing algorithms can be implemented so that they (1) produce drawings with all vertices placed at intersection points of an integer grid of size O(n 2 ), (2) perform arithmetic operations on integers only, and (3) run in O(n) time, where n is the number of vertices of the input graph. 1 Introduction A proximity drawing of a graph is a straight-line drawing (vertices are represented by points and edges by straight-line segments) where the points representing adjacent vertices are deemed to be close according t...
Generating outerplanar graphs uniformly at random
- IN COMBINATORICS, PROBABILITY, AND COMPUTATION
, 2003
"... We show how to generate labeled and unlabeled outerplanar graphs with n vertices uniformly at random in polynomial time in n. To generate labeled outerplanar graphs, we present a counting technique using the decomposition of a graph according to its block structure, and compute the exact number of ..."
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Cited by 9 (6 self)
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We show how to generate labeled and unlabeled outerplanar graphs with n vertices uniformly at random in polynomial time in n. To generate labeled outerplanar graphs, we present a counting technique using the decomposition of a graph according to its block structure, and compute the exact number of labeled outerplanar graphs. This allows us to make the correct probabilistic choices in a recursive generation of uniformly distributed outerplanar graphs. Next we modify our formulas to count rooted unlabeled graphs, and finally show how to use these formulas in a Las Vegas algorithm to generate unlabeled outerplanar graphs uniformly at random in expected polynomial time.
Witness Gabriel graphs
, 2009
"... We consider a generalization of the Gabriel graph, the witness Gabriel graph. Given a set of vertices P and a set of witness points W in the plane, there is an edge ab between two points of P in the witness Gabriel graph GG−(P,W) if and only if the closed disk with diameter ab does not contain any w ..."
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Cited by 8 (3 self)
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We consider a generalization of the Gabriel graph, the witness Gabriel graph. Given a set of vertices P and a set of witness points W in the plane, there is an edge ab between two points of P in the witness Gabriel graph GG−(P,W) if and only if the closed disk with diameter ab does not contain any witness point (besides possibly a and/or b). We study several properties of the witness Gabriel graph, both as a proximity graph and as a new tool in graph drawing. 1
Area Requirement of Gabriel Drawings
, 1996
"... In this paper we investigate the area requirement of proximity drawings and we prove an exponential lower bound. Namely, 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. Also, we extend ..."
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Cited by 6 (5 self)
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In this paper we investigate the area requirement of proximity drawings and we prove an exponential lower bound. Namely, 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. Also, we extend the result to an infinite class of proximity drawings.
Generating Random Outerplanar Graphs
- In 1 st Workshop on Algorithms for Listing, Counting, and Enumeration
, 2003
"... We show how to generate labeled and unlabeled outerplanar graphs with n vertices uniformly at random in (expected) polynomial time in n. To generate these graphs, we present a new counting technique using the decomposition of a graph according to its block structure and compute the exact number ..."
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Cited by 6 (2 self)
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We show how to generate labeled and unlabeled outerplanar graphs with n vertices uniformly at random in (expected) polynomial time in n. To generate these graphs, we present a new counting technique using the decomposition of a graph according to its block structure and compute the exact number of labeled outerplanar graphs. This allows us to make the correct probabilistic choices in a recursive generation of uniformly distributed outerplanar graphs.
Proximity drawings in polynomial area and volume
, 2004
"... 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 yi ..."
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Cited by 3 (0 self)
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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 (a) polynomial area weak Gabriel drawings of ternary trees, (b)polynomial area weak β-proximity drawing of binary trees for any 0 � β<∞, and(c)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 β-proximity drawing and a relative neighborhood drawing. All described algorithms run in linear time.
Drawable and Forbidden Minimum Weight Triangulations (Extended Abstract)
"... 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 twofo ..."
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Cited by 2 (0 self)
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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...
Proximity Drawings of Binary Trees in Polynomial Area
"... In this paper, we study weak β-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 ..."
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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.
