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On Representing Graphs by Touching Cuboids
"... We consider contact representations of graphs where vertices are represented by cuboids, i.e. interiordisjoint axisaligned boxes in 3D space. Edges are represented by a proper contact between the cuboids representing their endvertices. Two cuboids make a proper contact if they intersect and thei ..."
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We consider contact representations of graphs where vertices are represented by cuboids, i.e. interiordisjoint axisaligned boxes in 3D space. Edges are represented by a proper contact between the cuboids representing their endvertices. Two cuboids make a proper contact if they intersect and their intersection is a nonzero area rectangle contained in the boundary of both. We study representations where all cuboids are unit cubes, where they are cubes of different sizes, and where they are axisaligned 3D boxes. We prove that it is NPcomplete to decide whether a graph admits a proper contact representation by unit cubes. We also describe algorithms that compute proper contact representations of varying size cubes for relevant graph families. Finally, we give two new simple proofs of a theorem by Thomassen stating that all planar graphs have a proper contact representation by touching cuboids.
LinearTime Algorithms for Holefree Rectilinear Proportional Contact Graph Representations
"... 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 first study proportional contact representation ..."
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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 first 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 O(n log n) time. We describe a new algorithm that guarantees 10sided rectilinear polygons and runs in O(n) time. We also describe a lineartime algorithm for proportional contact representation of planar 3trees with 8sided rectilinear polygons and show that this is optimal, as there exist planar 3trees that require 8sided polygons. We then show that a maximal outerplanar graph admits a proportional contact representation using rectilinear polygons with 6 sides when the outerboundary is a rectangle and with 4 sides otherwise. Finally we study maximal seriesparallel graphs. Here we show that O(1)sided rectilinear polygons are not possible unless we allow holes, but 6sided polygons can be achieved with arbitrarily small holes.
Proportional Contact Representations of 4connected Planar Graphs
"... Abstract. In a contact representation of a planar graph, vertices are represented by interiordisjoint polygons and two polygons share a nonempty common boundary when the corresponding vertices are adjacent. In the weighted version, a weight is assigned to each vertex and a contact representation i ..."
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Abstract. In a contact representation of a planar graph, vertices are represented by interiordisjoint polygons and two polygons share a nonempty common boundary when the corresponding vertices are adjacent. In the weighted version, a weight is assigned to each vertex and a contact representation is called proportional if each polygon realizes an area proportional to the vertex weight. In this paper we study proportional contact representations of 4connected internally triangulated planar graphs. The best known lower and upper bounds on the polygonal complexity for such graphs are 4 and 8, respectively. We narrow the gap between them by proving the existence of a representation with complexity 6. We then disprove a 10year old conjecture on the existence of a Hamiltonian canonical cycle in a 4connected maximal planar graph, which also implies that a previously suggested method for constructing proportional contact representations of complexity 6 for these graphs will not work. Finally we prove that it is NPcomplete to decide whether a 4connected planar graph admits a contact representation using only rectangles. 1
Table Cartograms
"... A table cartogram of a two dimensional m × n table A of nonnegative weights in a rectangle R, whose area equals the sum of the weights, is a partition of R into convex quadrilateral faces corresponding to the cells of A such that each face has the same adjacency as its corresponding cell and has a ..."
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A table cartogram of a two dimensional m × n table A of nonnegative weights in a rectangle R, whose area equals the sum of the weights, is a partition of R into convex quadrilateral faces corresponding to the cells of A such that each face has the same adjacency as its corresponding cell and has area equal to the cell’s weight. Such a partition acts as a natural way to visualize table data arising in various fields of research. In this paper, we give a O(mn)time algorithm to find a table cartogram in a rectangle. We then generalize our algorithm to obtain table cartograms inside arbitrary convex quadrangles, circles, and finally, on the surface of cylinders and spheres.
The Order Dimension of Planar Maps Revisited
"... Schnyder characterized planar graphs in terms of order dimension. This seminal result found several extensions. A particularly far reaching extension is the BrightwellTrotter Theorem about planar maps. It states that the order dimension of the incidence poset PM of vertices, edges and faces of a ..."
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Schnyder characterized planar graphs in terms of order dimension. This seminal result found several extensions. A particularly far reaching extension is the BrightwellTrotter Theorem about planar maps. It states that the order dimension of the incidence poset PM of vertices, edges and faces of a planar map M has dimension at most 4. The original proof generalizes the machinery of Schnyderpaths and Schnyderregions. In this note we use a simple result about the order dimension of grid intersection graphs to show a slightly stronger result: dim(split(PM)) ≤ 4. This may be the first result in the area that is obtained without using the tools introduced by Schnyder.
Circulararc cartograms
"... We present a new circulararc cartogram model in which countries are drawn as polygons with circular arcs instead of straightline segments. Given a political map and values associated with each country in the map, a cartogram is a distorted map in which the areas of the countries are proportional ..."
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We present a new circulararc cartogram model in which countries are drawn as polygons with circular arcs instead of straightline segments. Given a political map and values associated with each country in the map, a cartogram is a distorted map in which the areas of the countries are proportional to the corresponding values. In the circulararc cartogram model straightline segments can be replaced by circular arcs in order to modify the areas of the polygons, while the corners of the polygons remain fixed. The countries in circulararc cartograms have the aesthetically pleasing appearance of clouds or snowflakes, depending on whether their edges are bent outwards or inwards. This makes it easy to determine whether a country has grown or shrunk, just by its overall shape. We show that determining whether a given map and given areavalues can be realized as a circulararc cartogram is an NPhard problem. Next we describe a heuristic method for constructing circulararc cartograms,
3D Proportional Contact Representations of Graphs
"... In 3D contact representations, the vertices of a graph are represented by 3D polyhedra and the edges are realized by nonzeroarea common boundaries between corresponding polyhedra. While contact representations with cuboids have been studied in the literature, we consider a novel generalization of ..."
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In 3D contact representations, the vertices of a graph are represented by 3D polyhedra and the edges are realized by nonzeroarea common boundaries between corresponding polyhedra. While contact representations with cuboids have been studied in the literature, we consider a novel generalization of the problem in which vertices are represented by axisaligned polyhedra that are union of two cuboids. In particular, we study the weighted (proportional) version of the problem, where the volumes of the polyhedra and the areas of the common boundaries realize prespecified vertex and edge weights. For some classes of graphs (e.g., outerplanar, planar bipartite, planar, complete), we provide algorithms to construct such representations for arbitrary given weights. We also show that not all graphs can be represented in 3D with axisaligned polyhedra of constant complexity.