Conference Paper
Triangle Contact Representations and Duality
DOI: 10.1007/9783642184697_24 Conference: Graph Drawing  18th International Symposium, GD 2010, Konstanz, Germany, September 2124, 2010. Revised Selected Papers
Source: DBLP
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 "The last section of the paper is dedicated to primaldual contact representations by triangles. We give a simple proof of a theorem of Gonçalves, Lévêque and Pinlou, which shows that every 3connected planar graph has a primaldual contact representation by triangles [13] "
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ABSTRACT: A straight line triangle representation (SLTR) of a planar graph is a straight line drawing such that all the faces including the outer face have triangular shape. Such a drawing can be viewed as a tiling of a triangle using triangles with the input graph as skeletal structure. In this paper we present a characterization of graphs that have an SLTR that is based on flat angle assignments, i.e., selections of angles of the graph that have size π in the representation. We also provide a second characterization in terms of contact systems of pseudosegments. With the aid of discrete harmonic functions we show that contact systems of pseudosegments that respect certain conditions are stretchable. The stretching procedure is then used to get straight line triangle representations. Since the discrete harmonic function approach is quite flexible it allows further applications, we mention some of them. The drawback of the characterization of SLTRs is that we are not able to effectively check whether a given graph admits a flat angle assignment that fulfills the conditions. Hence it is still open to decide whether the recognition of graphs that admit straight line triangle representation is polynomially tractable. 
 "Among their most prominent applications are the following: They provide a machinery to construct spaceefficient straightline drawings [23] [17] [7], yield a characterization of planar graphs via the dimension of their vertexedge incidence poset [22] [7], and are used to encode triangulations [21] [3]. Further applications lie in enumeration [4], representation by geometric objects [12] [15], graph spanners [5], etc. The richness of these applications has stimulated some research towards generalizing Schnyder woods to non planar graphs. "
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ABSTRACT: We propose a simple generalization of Schnyder woods from the plane to maps on orientable surfaces of higher genus. This is done in the language of angle labelings. Generalizing results of De Fraysseix and Ossona de Mendez, and Felsner, we establish a correspondence between these labelings and orientations and characterize the set of orientations of a map that correspond to such a Schnyder wood. Furthermore, we study the set of these orientations of a given map and provide a natural partition into distributive lattices depending on the surface homology. This generalizes earlier results of Felsner and Ossona de Mendez. In the toroidal case, a new proof for the existence of Schnyder woods is derived from this approach. 
 "In a contact representation of a planar graph, the vertices are represented by nonoverlapping geometric objects such as circles, polygons, or line segments and the edges are realized by a prespecified type of contact between these objects. Contact graphs of circles, made famous by the Koebe–Andreev–Thurston circle packing theorem [25], have many applications in graph drawing [1] [4] [5] [14] [15] [24] [28] [29] [35], and this success has motivated the study of many other contact representations [3] [12] [13] [18]. The special cases of contact representations with curves and in particular with line segments are of particular interest [7] [10] [11] [22] [23]. "
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ABSTRACT: We study representations of graphs by contacts of circular arcs, CCArepresentations for short, where the vertices are interiordisjoint circular arcs in the plane and each edge is realized by an endpoint of one arc touching the interior of another. A graph is (2,k)sparse if every svertex subgraph has at most 2s  k edges, and (2, k)tight if in addition it has exactly 2n  k edges, where n is the number of vertices. Every graph with a CCA representation is planar and (2, 0)sparse, and it follows from known results on contacts of line segments that for k >= 3 every (2, k)sparse graph has a CCArepresentation. Hence the question of CCArepresentability is open for (2, k)sparse graphs with 0 <= k <= 2. We partially answer this question by computing CCArepresentations for several subclasses of planar (2,0)sparse graphs. In particular, we show that every plane (2, 2)sparse graph has a CCArepresentation, and that any plane (2, 1)tight graph or (2, 0)tight graph dual to a (2, 3)tight graph or (2, 4)tight graph has a CCArepresentation. Next, we study CCArepresentations in which each arc has an empty convex hull. We characterize the plane graphs that have such a representation, based on the existence of a special orientation of the graph edges. Using this characterization, we show that every plane graph of maximum degree 4 has such a representation, but that finding such a representation for a plane (2, 0)tight graph with maximum degree 5 is an NPcomplete problem. Finally, we describe a simple algorithm for representing plane (2, 0)sparse graphs with wedges, where each vertex is represented with a sequence of two circular arcs (straightline segments).