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13
On rectilinear duals for vertexweighted plane graphs
 In GD ’05: Proceedings of the Symposium on Graph Drawing
, 2005
"... Abstract. Let G =(V,E) be a plane triangulated graph where each vertex is assigned a positive weight. A rectilinear dual of G is a partition of a rectangle into V  simple rectilinear regions, one for each vertex, such that two regions are adjacent if and only if the corresponding vertices are conn ..."
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Cited by 18 (1 self)
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Abstract. Let G =(V,E) be a plane triangulated graph where each vertex is assigned a positive weight. A rectilinear dual of G is a partition of a rectangle into V  simple rectilinear regions, one for each vertex, such that two regions are adjacent if and only if the corresponding vertices are connected by an edge in E. A rectilinear dual is called a cartogram if the area of each region is equal to the weight of the corresponding vertex. We show that every vertexweighted plane triangulated graph G admits a cartogram of constant complexity, that is, a cartogram where the number of vertices of each region is constant. 1
GMap: Visualizing Graphs and Clusters as Maps
, 2009
"... Information visualization is essential in making sense out of large data sets. Often, highdimensional data are visualized as a collection of points in 2dimensional space through dimensionality reduction techniques. However, these traditional methods often do not capture well the underlying structu ..."
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Cited by 12 (9 self)
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Information visualization is essential in making sense out of large data sets. Often, highdimensional data are visualized as a collection of points in 2dimensional space through dimensionality reduction techniques. However, these traditional methods often do not capture well the underlying structural information, clustering, and neighborhoods. In this paper, we describe GMap, a practical tool for visualizing relational data with geographiclike maps. We illustrate the effectiveness of this approach with examples from several domains. All the maps referenced in this paper can be found in www.research.att.com/˜yifanhu/GMap.
Orderly Spanning Trees with Applications
 SIAM Journal on Computing
, 2005
"... Abstract. We introduce and study orderly spanning trees of plane graphs. This algorithmic tool generalizes canonical orderings, which exist only for triconnected plane graphs. Although not every plane graph admits an orderly spanning tree, we provide an algorithm to compute an orderly pair for any c ..."
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Cited by 12 (1 self)
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Abstract. We introduce and study orderly spanning trees of plane graphs. This algorithmic tool generalizes canonical orderings, which exist only for triconnected plane graphs. Although not every plane graph admits an orderly spanning tree, we provide an algorithm to compute an orderly pair for any connected planar graph G, consisting of an embedded planar graph H isomorphic to G, and an orderly spanning tree of H. We also present several applications of orderly spanning trees: (1) a new constructive proof for Schnyder’s realizer theorem, (2) the first algorithm for computing an areaoptimal 2visibility drawing of a planar graph, and (3) the most compact known encoding of a planar graph with O(1)time query support. All algorithms in this paper run in linear time.
Optimal Polygonal Representation of Planar Graphs
"... Abstract. In this paper, we consider the problem of representing graphs by polygons whose sides touch. We show that at least six sides per polygon are necessary by constructing a class of planar graphs that cannot be represented by pentagons. We also show that the lower bound of six sides is matched ..."
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Cited by 10 (9 self)
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Abstract. In this paper, we consider the problem of representing graphs by polygons whose sides touch. We show that at least six sides per polygon are necessary by constructing a class of planar graphs that cannot be represented by pentagons. We also show that the lower bound of six sides is matched by an upper bound of six sides with a linear time algorithm for representing any planar graph by touching hexagons. Moreover, our algorithm produces convex polygons with edges with slopes 0, 1,1. 1
Computing Cartograms with Optimal Complexity
"... In a rectilinear dual of a planar graph vertices are represented by simple rectilinear polygons and edges are represented by sidecontact between the corresponding polygons. A rectilinear dual is called a cartogram if the area of each region is equal to a prespecified weight of the corresponding ve ..."
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Cited by 9 (7 self)
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In a rectilinear dual of a planar graph vertices are represented by simple rectilinear polygons and edges are represented by sidecontact between the corresponding polygons. A rectilinear dual is called a cartogram if the area of each region is equal to a prespecified weight of the corresponding vertex. The complexity of a cartogram is determined by the maximum number of corners (or sides) required for any polygon. In a series of papers the polygonal complexity of such representations for maximal planar graphs has been reduced from the initial 40 to 34, then to 12 and very recently to the currently best known 10. Here we describe a construction with 8sided polygons, which is optimal in terms of polygonal complexity as 8sided polygons are sometimes necessary. Specifically, we show how to compute the combinatorial structure and how to refine the representation into an areauniversal rectangular layout in linear time. The exact cartogram can be computed from the areauniversal rectangular layout with numerical iteration, or can be approximated with a hillclimbing heuristic. We also describe an alternative construction for Hamiltonian maximal planar graphs, which allows us to directly compute the cartograms in linear time. Moreover, we prove that even for Hamiltonian graphs 8sided rectilinear polygons are necessary, by constructing a nontrivial lower bound example. The complexity of the cartograms can be reduced to 6 if the Hamiltonian path has the extra property that it is onelegged, as in outerplanar graphs. Thus, we have optimal representations (in terms of both polygonal complexity and running time) for Hamiltonian maximal planar and maximal outerplanar graphs.
Areauniversal rectangular layouts
 In Proc. 25th ACM Symp. Computational Geometry
, 2009
"... Abstract. We construct partitions of rectangles into smaller rectangles from an input consisting of a planar dual graph of the layout together with restrictions on the orientations of edges and junctions of the layout. Such an orientationconstrained layout, if it exists, may be constructed in polyno ..."
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Cited by 7 (2 self)
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Abstract. We construct partitions of rectangles into smaller rectangles from an input consisting of a planar dual graph of the layout together with restrictions on the orientations of edges and junctions of the layout. Such an orientationconstrained layout, if it exists, may be constructed in polynomial time, and all orientationconstrained layouts may be listed in polynomial time per layout. 1
On Touching Triangle Graphs
"... Abstract. In this paper, we consider the problem of representing graphs by triangles whose sides touch. We present linear time algorithms for creating touching triangles representations for outerplanar graphs, square grid graphs, and hexagonal grid graphs. The class of graphs with touching triangles ..."
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Cited by 5 (3 self)
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Abstract. In this paper, we consider the problem of representing graphs by triangles whose sides touch. We present linear time algorithms for creating touching triangles representations for outerplanar graphs, square grid graphs, and hexagonal grid graphs. The class of graphs with touching triangles representations is not closed under minors, making characterization difficult. We do show that pairs of vertices can only have a small common neighborhood, and we present a complete characterization of the subclass of biconnected graphs that can be represented as triangulations of some polygon.
Rectangle and Square Representations of Planar Graphs
"... In the first part of this survey we consider planar graphs that can be represented by a dissections of a rectangle into rectangles. In rectangular drawings the corners of the rectangles represent the vertices. The graph obtained by taking the rectangles as vertices and contacts as edges is the recta ..."
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Cited by 5 (3 self)
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In the first part of this survey we consider planar graphs that can be represented by a dissections of a rectangle into rectangles. In rectangular drawings the corners of the rectangles represent the vertices. The graph obtained by taking the rectangles as vertices and contacts as edges is the rectangular dual. In visibility graphs and segment contact graphs the vertices correspond to horizontal or to horizontal and vertical segments of the dissection. Special orientations of graphs turn out to be helpful when dealing with characterization and representation questions. Therefore, we look at orientations with prescribed degrees, bipolar orientations, separating decompositions, and transversal structures. In the second part we ask for representations by a dissections of a rectangle into squares. We
Lineartime algorithms for proportional contact graph representations
, 2011
"... Abstract. In a proportional contact representation of a planar graph, each vertex is represented by a simple polygon with area proportional to a given weight, and edges are represented by adjacencies between the corresponding pairs of polygons. In this paper we study proportional contact representat ..."
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Cited by 3 (2 self)
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Abstract. In a proportional contact representation of a planar graph, each vertex is represented by a simple polygon with area proportional to a given weight, and edges are represented by adjacencies between the corresponding pairs of polygons. In this paper we study proportional contact representations that use rectilinear polygons without wasted areas (white space). In this setting, the best known algorithm for proportional contact representation of a maximal planar graph uses 12sided rectilinear polygons and takes�time. We describe a new algorithm that guarantees 10sided rectilinear polygons and runs in time. We also describe a lineartime algorithm for proportional contact representation of planar 3trees with 8sided rectilinear polygons and show that this optimal, as there exist planar 3trees that requires 8sided polygons. Finally, we show that a maximal outerplanar graph admits a proportional contact representation with 6sided rectilinear polygons when the outerboundary is a rectangle and with 4 sides otherwise. 1
Regular Labelings and Geometric Structures
, 2010
"... Three types of geometric structure—grid triangulations, rectangular subdivisions, and orthogonal polyhedra— can each be described combinatorially by a regular labeling: an assignment of colors and orientations to the edges of an associated maximal or nearmaximal planar graph. We briefly survey the ..."
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Cited by 2 (1 self)
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Three types of geometric structure—grid triangulations, rectangular subdivisions, and orthogonal polyhedra— can each be described combinatorially by a regular labeling: an assignment of colors and orientations to the edges of an associated maximal or nearmaximal planar graph. We briefly survey the connections and analogies between these three kinds of labelings, and their uses in designing efficient geometric algorithms.