Optimal Polygonal Representation of Planar Graphs

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

doi:10.1007/978-3-642-12200-2_37

Cited by 12 publications

(13 citation statements)

References 29 publications

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“…Can this be extended to allow for holes? Finally, the work on hexagonal contact graphs [10] gives a small (|V | × |V |) area bound. Are small areas possible in the triangular or, at least, the outerplanar case?…”

Section: Discussionmentioning

confidence: 99%

“…Kant's linear time algorithm for drawing degree-3 planar graphs on a hexagonal grid [15] can be used to obtain hexagonal drawings for planar graphs. Gansner et al [10] show that at least six sides are necessary and that the lower bound is matched by an upper bound of six sides with a linear time algorithm for representing any planar graph by touching convex hexagons.…”

Section: Related Workmentioning

confidence: 99%

“…Gansner et al [10] have shown that six sides are not only sufficient but also necessary, and gave a linear time construction. This leads us to consider which planar graphs can be represented by polygons with fewer than six sides.…”

Section: Introductionmentioning

confidence: 99%

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On Touching Triangle Graphs

Gansner

Kobourov

2011

Graph Drawing

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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.

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Section: Discussionmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

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On Touching Triangle Graphs

Gansner

Kobourov

2011

Graph Drawing

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“…Moreover, they show that 6 sides are necessary if convexity is required. For maximal planar graphs, the representation obtained by the algorithm in [9] is hole-free. Buchsbaum et al [4] give an overview on the state of the art concerning rectangle contact graphs.…”

Section: Related Workmentioning

confidence: 99%

“…While the above results deal with point-contacts, the problem of constructing sidecontact representations is less studied. Gansner et al [9] show that any planar graph G has a side-contact representation with convex hexagons. Moreover, they show that 6 sides are necessary if convexity is required.…”

Section: Related Workmentioning

confidence: 99%

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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Triangle Contact Representations and Duality

Lévêque

Pinlou

2011

Graph Drawing

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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 representation 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 corresponding 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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Optimal Polygonal Representation of Planar Graphs

Cited by 12 publications

References 29 publications

On Touching Triangle Graphs

On Touching Triangle Graphs

Proportional Contact Representations of Planar Graphs

Triangle Contact Representations and Duality

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