No-Bend Orthogonal Drawings and No-Bend Orthogonally Convex Drawings of Planar Graphs (Extended Abstract)

Hasan, Md. Manzurul; Rahman, Md. Saidur

doi:10.1007/978-3-030-26176-4_21

Cited by 12 publications

(5 citation statements)

References 11 publications

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“…Further, every efficient algorithm for testing the existence of planar rectilinear drawings has been eventually subsumed by an algorithm in the more general bend-minimization scenario. This was indeed the case for degree-3 series-parallel graphs with a variable embedding (see [18] and [24]), for degree-3 planar graphs with a fixed embedding (see [20] and [19]), and for degree-3 planar graphs with a variable embedding (see [13] and [8]). Figures 1(a) and 1(b) show two planar rectilinear drawings.…”

Section: Introductionmentioning

confidence: 61%

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Planar rectilinear drawings of outerplanar graphs in linear time

Frati¹

2022

Computational Geometry

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Section: Introductionmentioning

confidence: 61%

“…O(n) [20] O(n 1.5 ) [4] O(n) [13] NP-hard [11] Series-Parallel O(n) [20] O(n 1.5 ) [4] O(n) [18] O(n 3 log n) [7] Outerplanar O(n) [20] O(n) This paper O(n) [17] O(n) This paper Table 1. Complexity of testing for the existence of a planar rectilinear drawing.…”

Section: Planarmentioning

confidence: 99%

Planar rectilinear drawings of outerplanar graphs in linear time

Frati¹

2022

Computational Geometry

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“…Rectilinear planarity testing is NP-complete for planar 4-graphs [16], but it is polynomial-time solvable for planar 3-graphs [5,9] and linear-time solvable for subdivisions of planar triconnected cubic graphs [23]. Very recently a linear-time algorithm for rectilinear planarity testing of biconnected planar 3-graphs has been presented [18]. By extending a result of Thomassen [29] about 3-graphs that have a rectilinear drawing with all rectangular faces, Rahman et al [25] characterize rectilinear plane 3-graphs (see Theorem 2.1).…”

Section: Preliminariesmentioning

confidence: 99%

Optimal Orthogonal Drawings of Planar 3-Graphs in Linear Time

Didimo

Liotta

Ortali

et al. 2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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A planar orthogonal drawing Γ of a planar graph G is a geometric representation of G such that the vertices are drawn as distinct points of the plane, the edges are drawn as chains of horizontal and vertical segments, and no two edges intersect except at their common endpoints. A bend of Γ is a point of an edge where a horizontal and a vertical segment meet. Γ is bend-minimum if it has the minimum number of bends over all possible planar orthogonal drawings of G. This paper addresses a long standing, widely studied, open question: Given a planar 3-graph G (i.e., a planar graph with vertex degree at most three), what is the best computational upper bound to compute a bend-minimum planar orthogonal drawing of G in the variable embedding setting? In this setting the algorithm can choose among the exponentially many planar embeddings of G the one that leads to an orthogonal drawing with the minimum number of bends. We answer the question by describing an O(n)time algorithm that computes a bend-minimum planar orthogonal drawing of G with at most one bend per edge, where n is the number of vertices of G. The existence of an orthogonal drawing algorithm that simultaneously minimizes the total number of bends and the number of bends per edge was previously unknown.

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“…Namely, if the algorithm must preserve a given planar embedding, rectilinear planarity testing can be solved in subquadratic time for general graphs [2,13], and in linear time for planar 3-graphs [22] and for biconnected series-parallel graphs (SP-graphs for short) [7]. When the planar embedding is not fixed, linear-time solutions exist for (families of) planar 3-graphs [10,16,21,24] and for outerplanar graphs [12]. A polynomial-time solution for SP-graphs has been known for a long time [5], but establishing whether there is a linear-time algorithm for this graph family remains a long-standing open problem [1]; to date, the most efficient algorithm for n-vertex SP-graphs has complexity O(n 3 log n) [6].…”

Section: Introductionmentioning

confidence: 99%

Spirality and Rectilinear Planarity Testing of Independent-Parallel SP-Graphs

Didimo,

Kaufmann,

Liotta

et al. 2021

Preprint

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We study the long-standing open problem of efficiently testing rectilinear planarity of series-parallel graphs (SP-graphs) in the variable embedding setting. A key ingredient behind the design of a lineartime testing algorithm for SP-graphs of vertex-degree at most three is that one can restrict the attention to a constant number of "rectilinear shapes" for each series or parallel component. To formally describe these shapes the notion of spirality can be used. This key ingredient no longer holds for SP-graphs with vertices of degree four, as we prove a logarithmic lower bound on the spirality of their components. The bound holds even for the independent-parallel SP-graphs, in which no two parallel components share a pole. Nonetheless, by studying the spirality properties of the independent-parallel SP-graphs, we are able to design a linear-time rectilinear planarity testing algorithm for this graph family.

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No-Bend Orthogonal Drawings and No-Bend Orthogonally Convex Drawings of Planar Graphs (Extended Abstract)

Cited by 12 publications

References 11 publications

Planar rectilinear drawings of outerplanar graphs in linear time

Planar rectilinear drawings of outerplanar graphs in linear time

Optimal Orthogonal Drawings of Planar 3-Graphs in Linear Time

Spirality and Rectilinear Planarity Testing of Independent-Parallel SP-Graphs

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