A linear time algorithm to check for the existence of a rectangular dual of a planar triangulated graph

Bhasker, J.; Sahni, Sartaj

doi:10.1002/net.3230170306

Cited by 58 publications

(48 citation statements)

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“…Not every plane graph has a rectangular dual. A plane graph G has a rectangular dual R with four rectangles on the boundary of R if G is an irreducible triangulation: (i) G is triangulated and the exterior face is a quadrangle; (ii) G has no separating triangles (a 3-cycle with vertices both inside and outside the cycle) [6,20]. A plane triangulated graph G has a rectangular dual if and only if we can augment G with four external vertices such that the augmented graph is an irreducible triangulation.…”

Section: Rectangular Partitionsmentioning

confidence: 99%

Optimizing Regular Edge Labelings

2011

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DisclaimerThis document contains a student thesis (bachelor's or master's), as authored by a student at Eindhoven University of Technology. Student theses are made available in the TU/e repository upon obtaining the required degree. The grade received is not published on the document as presented in the repository. The required complexity or quality of research of student theses may vary by program, and the required minimum study period may vary in duration. General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.• Users may download and print one copy of any publication from the public portal for the purpose of private study or research.• You may not further distribute the material or use it for any profit-making activity or commercial gain Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. AbstractA regular edge labeling of an irreducible triangulation G uniquely defines a rectangular dual of G. Rectangular duals find applications in various areas: as floor plans of electronic chips, in architectural designs, as rectangular cartograms, or as treemaps. An irreducible triangulation can have many regular edge labelings and hence many rectangular duals. Depending on the specific application different duals might be desirable. In this thesis we consider optimization problems on regular edge labelings and show how to find optimal or near-optimal ones for various quality criteria. We evaluate our optimization methods by applying them to generate high quality rectangular cartograms. Furthermore, we show how to efficiently enumerate the regular edge labelings of an irreducible triangulation. Since the running time of the enumeration algorithm depends on the number of regular edge labelings, we also consider the maximal number of regular edge labelings an irreducible triangulation can have. We show that every irreducible triangulation with n vertices has less than O(4.6807 n ) regular edge labelings and that there are irreducible triangulations with Ω(3.0426 n ) regular edge labelings.i

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Section: Rectangular Partitionsmentioning

confidence: 99%

Optimizing Regular Edge Labelings

2011

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“…Ungar [20], Bhasker and Sahni [4], and Koźmiński and Kinnen [12] independently gave equivalent characterizations of the rectangular graphs. Eppstein et al [8] characterized the area-universal rectangular duals.…”

Section: Introductionmentioning

confidence: 99%

“…An irreducible triangulation is a plane graph without separating triangles and where all interior faces are triangles and the outer face is a quadrangle. A graph G has a rectangular dual if and only if G has an extended graph which is an irreducible triangulation [4,12,20].…”

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Towards Characterizing Graphs with a Sliceable Rectangular Dual

Kusters

Speckmann

2015

Lecture Notes in Computer Science

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Abstract. Let G be a plane triangulated graph. A rectangular dual of G is a partition of a rectangle R into a set R of interior-disjoint rectangles, one for each vertex, such that two regions are adjacent if and only if the corresponding vertices are connected by an edge. A rectangular dual is sliceable if it can be recursively subdivided along horizontal or vertical lines. A graph is rectangular if it has a rectangular dual and sliceable if it has a sliceable rectangular dual. There is a clear characterization of rectangular graphs. However, a full characterization of sliceable graphs is still lacking. The currently best result (Yeap and Sarrafzadeh, 1995) proves that all rectangular graphs without a separating 4-cycle are sliceable. In this paper we introduce a recursively defined class of graphs and prove that these graphs are precisely the nonsliceable graphs with exactly one separating 4-cycle.

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“…Notice that this differs significantly from the papers mentioned above, where the objective is to find a minimum set of augmenting edges to reach a certain connectivity constraint. During the last few years 4-connected planar graphs received new attention due to their important characteristics: every 4-connected planar graph is hamiltonian, it can be drawn as a visibility representation in a very compact way [19], and if it is triangular it can be represented by a rectangular dual [1], [20]. Visibility representations and rectangular duals are widely used drawing representations, e.g., in industrial environments where rectangular duals are used in floor-planning problems [21].…”

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On Triangulating Planar Graphs under the Four-Connectivity Constraint

Biedl¹,

Kant²,

Kaufmann³

1997

Algorithmica

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Triangulation of planar graphs under constraints is a fundamental problem in the representation of objects. Related keywords are graph augmentation from the field of graph algorithms and mesh generation from the field of computational geometry. We consider the triangulation problem for planar graphs under the constraint to satisfy 4-connectivity. A 4-connected planar graph has no separating triangles, i.e., cycles of length 3 which are not a face.We show that triangulating embedded planar graphs without introducing new separating triangles can be solved in linear time and space. If the initial graph had no separating triangle, the resulting triangulation is 4-connected. If the planar graph is not embedded, then deciding whether there exists an embedding with at most k separating triangles is NP-complete. For biconnected graphs a linear-time approximation which produces an embedding with at most twice the optimal number is presented. With this algorithm we can check in linear time whether a biconnected planar graph can be made 4-connected while maintaining planarity. Several related remarks and results are included. Introduction. The problem of augmenting a graph to reach a certain connectivity requirement by adding edges has important applications in network reliability [9], [25]and fault-tolerant computing. The general version of the augmentation problem is to augment the input graph to reach a given connectivity requirement by adding a smallest set of edges. Recent papers present linear-time augmentation algorithms to admit the 2-connectivity constraint [8], [24], [15], and the 3-connectivity constraint [14]. With respect to 4-connectivity Kanevsky et al. [16] presented an O(n · α(m, n) + m)-time algorithm for testing 4-connectivity, and Hsu presented an O(n · α(m, n) + m)-time algorithm to compute the minimal set of edges to augment a 3-connected graph to a 4-connected graph [13] (here α(m, n) is the functional inverse of Ackermann's function). Kant described several algorithms for the augmentation problem with the additional constraint of planarity [18]. This problem has important applications in planar network design and graph-drawing algorithms.

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A linear time algorithm to check for the existence of a rectangular dual of a planar triangulated graph

Abstract: We develop a linear time algorithm to determine if a given planar triangulated graph has a rectangular dual.

Cited by 58 publications

References 9 publications

Optimizing Regular Edge Labelings

Optimizing Regular Edge Labelings

Towards Characterizing Graphs with a Sliceable Rectangular Dual

On Triangulating Planar Graphs under the Four-Connectivity Constraint

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