On Touching Triangle Graphs

Gansner, Emden R.; Hu, Yifan; Kobourov, Stephen G.

doi:10.1007/978-3-642-18469-7_23

Cited by 19 publications

(17 citation statements)

References 23 publications

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“…To establish necessity, consider the 2-tree obtained from K 2,4 by adding an edge between the vertices of the partition of size two. These two vertices have four common neighbors, but as was proved in [10], in any side-contact representation with triangles, any pair of adjacent vertices has at most three common neighbors. Hence this graph has no sidecontact representation with triangles, let alone one that respects the weights.…”

Section: Theorem 5 Let G = (V E) Be a 2-shellable Graph And W : V →mentioning

confidence: 83%

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Proportional Contact Representations of Planar Graphs

et al. 2012

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Abstract. We study contact representations for planar graphs, with vertices represented by simple polygons and adjacencies represented by point-contacts or side-contacts between the corresponding polygons. Specifically, we consider proportional contact representations, where pre-specified vertex weights must be represented by the areas of the corresponding polygons. Several natural optimization goals for such representations include minimizing the complexity of the polygons, the cartographic error, and the unused area. We describe constructive algorithms for proportional contact representations with optimal complexity for general planar graphs and planar 2-segment graphs, which include maximal outerplanar graphs and partial 2-trees.

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Section: Theorem 5 Let G = (V E) Be a 2-shellable Graph And W : V →mentioning

confidence: 83%

“…The characterization of graphs admitting a hole-free side-contact representation with rectangles was obtained by Kozḿiński and Kinnen [16] or in the dual setting by Ungar [19]. There is a also a simple linear time algorithm for constructing triangle side-contact representations for outerplanar graphs [10].…”

Section: Related Workmentioning

confidence: 99%

Proportional Contact Representations of Planar Graphs

et al. 2012

Self Cite

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show abstract

“…To establish necessity, consider the 2-tree obtained from K 2,4 by adding an edge between the vertices of the partition of size two. These two vertices have four common neighbors, but as was proved in [11], in any side-contact representation with triangles, any pair of adjacent vertices has at most three common neighbors. Hence this graph has no sidecontact representation with triangles, let alone one that respects the weights.…”

Section: Theorem 6 Four-sided Convex Polygons Are Always Sufficient mentioning

confidence: 83%

“…The characterization of graphs admitting a hole-free side-contact representation with rectangles was obtained by Kozḿiński and Kinnen [20] or in the dual setting by Ungar [24]. There is a also a simple linear time algorithm for constructing triangle side-contact representations for outerplanar graphs [11].…”

Section: Related Workmentioning

confidence: 99%

Proportional Contact Representations of Planar Graphs

Alam¹,

Biedl²,

Felsner³

et al. 2012

JGAA

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Abstract. We study contact representations for planar graphs, with vertices represented by simple polygons and adjacencies represented by a point-contact or a side-contact between the corresponding polygons. Specifically, we consider proportional contact representations, where pre-specified vertex weights must be represented by the areas of the corresponding polygons. Several natural optimization goals for such representations include minimizing the complexity of the polygons, the cartographic error, and the unused area. We describe constructive algorithms for proportional contact representations with optimal complexity for general planar graphs and planar 2-segment graphs, which include maximal outerplanar graphs and partial 2-trees.

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“…Gansner, Hu and Kobourov show that outerplanar graphs, grid graphs and hexagonal grid graphs are Touching Triangle Graphs (TTGs). They give a linear time algorithm to find the TTG [12]. Alam,Fowler and Kobourov [3] consider proper TTGs, i.e., the union of all triangles of the TTG is a triangle and there are no holes.…”

Section: Introductionmentioning

confidence: 99%

Straight Line Triangle Representations

Aerts

Felsner

2017

Discrete Comput Geom

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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. The characterization 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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On Touching Triangle Graphs

Cited by 19 publications

References 23 publications

Proportional Contact Representations of Planar Graphs

Proportional Contact Representations of Planar Graphs

Proportional Contact Representations of Planar Graphs

Straight Line Triangle Representations

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