An algorithm for 1-bend embeddings of plane graphs in the two-dimensional grid

Morgana, A.; Mello, Célia Picinin de; Sontacchi, Giovanna

doi:10.1016/s0166-218x(03)00373-1

Cited by 7 publications

(3 citation statements)

References 3 publications

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“…Unfortunately, their paper does not include any proofs and to the best of our knowledge a proof of these results did not appear. Morgana et al [11] characterize the class of plane graphs (i.e., planar graphs with a given embedding) that admit a 1-bend embedding on the grid by forbidden configurations. They also present a quadratic algorithm that either detects a forbidden configuration or computes a 1-bend embedding.…”

Section: Introductionmentioning

confidence: 99%

Orthogonal Graph Drawing with Flexibility Constraints

Bläsius

Krug

Rutter

et al. 2011

Lecture Notes in Computer Science

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In this work we consider the following problem. Given a planar graph G with maximum degree 4 and a function flex : E −→ N 0 that gives each edge a flexibility. Does G admit a planar embedding on the grid such that each edge e has at most flex(e) bends? Note that in our setting the combinatorial embedding of G is not fixed.We give a polynomial-time algorithm for this problem when the flexibility of each edge is positive. This includes as a special case the problem of deciding whether G admits a drawing with at most one bend per edge.

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Section: Introductionmentioning

confidence: 99%

Orthogonal Graph Drawing with Flexibility Constraints

Bläsius

Krug

Rutter

et al. 2011

Lecture Notes in Computer Science

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show abstract

“…In [17] forbidden configurations for the 1-embedding of a graph are characterized and an algorithm is provided that either detects a forbidden configuration or generates a 1-rectilinear embedding, in O(n 2 ) time.…”

Section: Embedding Of Plane Graphsmentioning

confidence: 99%

A tribute to Célia P. de Mello

Morgana¹,

Almeida²

2022

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“…Biedl and Kant [1] show that every plane graph can be embedded with at most two bends per edge except for the octahedron. Morgana et al [12] give a characterization of plane graphs that have an orthogonal drawing with at most one bend per edge. Tayu et al [17] show that every series-parallel graph can be drawn with at most one bend per edge.…”

Section: Introductionmentioning

confidence: 99%

Optimal Orthogonal Graph Drawing with Convex Bend Costs

Bläsius

Rutter

Wagner

2016

ACM Trans. Algorithms

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Traditionally, the quality of orthogonal planar drawings is quantified by the total number of bends or the maximum number of bends per edge. However, this neglects that, in typical applications, edges have varying importance. We consider the problem OPTIMALFLEXDRAW that is defined as follows. Given a planar graph G on n vertices with maximum degree 4 (4-planar graph) and for each edge e a cost function cost e : N 0 −→ R defining costs depending on the number of bends e has, compute a planar orthogonal drawing of G of minimum cost.In this generality OPTIMALFLEXDRAW is NP-hard. We show that it can be solved efficiently if (1) the cost function of each edge is convex and (2) the first bend on each edge does not cause any cost. Our algorithm takes time O(n · T flow (n)) and O(n 2 · T flow (n)) for biconnected and connected graphs, respectively, where T flow (n) denotes the time to compute a minimum-cost flow in a planar network with multiple sources and sinks. Our result is the first polynomial-time bend-optimization algorithm for general 4-planar graphs optimizing over all embeddings. Previous work considers restricted graph classes and unit costs.

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An algorithm for 1-bend embeddings of plane graphs in the two-dimensional grid

Cited by 7 publications

References 3 publications

Orthogonal Graph Drawing with Flexibility Constraints

Orthogonal Graph Drawing with Flexibility Constraints

A tribute to Célia P. de Mello

Optimal Orthogonal Graph Drawing with Convex Bend Costs

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