Conference Paper

# Triangle Contact Representations and Duality

DOI: 10.1007/978-3-642-18469-7_24 Conference: Graph Drawing - 18th International Symposium, GD 2010, Konstanz, Germany, September 21-24, 2010. Revised Selected Papers
Source: DBLP

ABSTRACT

A contact representation by triangles of a graph is a set of triangles in the plane such that two triangles intersect on at most one point, each triangle represents a vertex of the graph and two triangles intersects if and only if their corresponding vertices are adjacent. de Fraysseix, Ossona de Mendez and Rosenstiehl proved that every planar graph admits a contact representation by triangles. We strengthen this in terms of a simultaneous contact representation by triangles of a planar map and of its dual. A primal-dual contact representation by triangles of a planar map is a contact representa- tion by triangles of the primal and a contact representation by triangles of the dual such that for every edge uv, bordering faces f and g, the intersection between the triangles corresponding to u and v is the same point as the intersection between the triangles cor- responding to f and g. We prove that every 3-connected planar map admits a primal-dual contact representation by triangles. Moreover, the interiors of the triangles form a tiling of the triangle corresponding to the outer face and each contact point is a node of exactly three triangles. Then we show that these representations are in one-to-one correspondence with generalized Schnyder woods defined by Felsner for 3-connected planar maps.

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• "The last section of the paper is dedicated to primal-dual contact representations by triangles. We give a simple proof of a theorem of Gonçalves, Lévêque and Pinlou, which shows that every 3-connected planar graph has a primal-dual contact representation by triangles [13] "
##### Article: Straight Line Triangle Representations
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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.
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• "Among their most prominent applications are the following: They provide a machinery to construct space-efficient straight-line drawings [23] [17] [7], yield a characterization of planar graphs via the dimension of their vertex-edge 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. "
##### Article: Structure of Schnyder labelings on orientable surfaces
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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.
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• "In a contact representation of a planar graph, the vertices are represented by non-overlapping 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]. "
##### Article: Contact Representations of Sparse Planar Graphs
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ABSTRACT: We study representations of graphs by contacts of circular arcs, CCA-representations for short, where the vertices are interior-disjoint 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 s-vertex 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 CCA-representation. Hence the question of CCA-representability is open for (2, k)-sparse graphs with 0 <= k <= 2. We partially answer this question by computing CCA-representations for several subclasses of planar (2,0)-sparse graphs. In particular, we show that every plane (2, 2)-sparse graph has a CCA-representation, 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 CCA-representation. Next, we study CCA-representations 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 NP-complete 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 (straight-line segments).