Planar L-Drawings of Directed Graphs

Chaplick, Steven; Chimani, Markus; Cornelsen, Sabine; Lozzo, Giordano Da; Nöllenburg, Martin; Patrignani, Maurizio; Tollis, Ioannis G.; Wolff, Alexander

doi:10.1007/978-3-319-73915-1_36

Cited by 14 publications

(16 citation statements)

References 17 publications

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“…An L-drawing realizes a port assignment if each edge e = (v, w) is incident to the out(e)-port of v and to the in(e)-port of w. A port assignment admits a planar L-drawing if there is a planar L-drawing that realizes it. Given a port assignment it can be tested in linear time whether it admits a planar L-drawing [10].…”

Section: Preliminariesmentioning

confidence: 99%

“…Non-planar L-drawings were first defined by Angelini et al [1]. Chaplick et al [10] showed that it is NP-complete to decide whether a directed graph has a planar L-drawing if the embedding is not fixed. However it can be decided in linear time whether a planar st-graph has an upward-planar L-drawing, i.e.…”

Section: Introductionmentioning

confidence: 99%

“…Each undirected simple graph admits a Kandinsky drawing with one bend per edge [9]. The relationship between Kandinsky drawings and planar L-drawings was established in [10].…”

Section: Introductionmentioning

confidence: 99%

“…This implies that every upward-planar graph admits a planar L-drawing respecting a given upward-planar embedding. We thus solve an open problem posed in [10]. The construction is based on rectangular duals.…”

Section: Introductionmentioning

confidence: 99%

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Planar L-Drawings of Bimodal Graphs

Angelini¹,

Chaplick²,

Cornelsen³

et al. 2020

Preprint

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In a planar L-drawing of a directed graph (digraph) each edge e is represented as a polyline composed of a vertical segment starting at the tail of e and a horizontal segment ending at the head of e. Distinct edges may overlap, but not cross. Our main focus is on bimodal graphs, i.e., digraphs admitting a planar embedding in which the incoming and outgoing edges around each vertex are contiguous. We show that every plane bimodal graph without 2-cycles admits a planar L-drawing. This includes the class of upward-plane graphs. Finally, outerplanar digraphs admit a planar L-drawing -although they do not always have a bimodal embedding -but not necessarily with an outerplanar embedding.

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Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Each undirected simple graph admits a Kandinsky drawing with one bend per edge [9]. The relationship between Kandinsky drawings and planar L-drawings was established in [10].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Planar L-Drawings of Bimodal Graphs

Angelini¹,

Chaplick²,

Cornelsen³

et al. 2020

Preprint

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“…A planar st-graph G is a bitonic st-graph if it admits a bitonic st-ordering. Deciding whether G is bitonic can be done in linear time both in the fixed [18] and in the variable [7] embedding settings. If G is not bitonic, every st-ordering σ of G contains a forbidden configuration defined as follows.…”

Section: Preliminariesmentioning

confidence: 99%

Universal Slope Sets for Upward Planar Drawings

Bekos¹,

Giacomo²,

Didimo³

et al. 2018

Lecture Notes in Computer Science

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We prove that every set S of ∆ slopes containing the horizontal slope is universal for 1-bend upward planar drawings of bitonic st-graphs with maximum vertex degree ∆, i.e., every such digraph admits a 1-bend upward planar drawing whose edge segments use only slopes in S. This result is worst-case optimal in terms of the number of slopes, and, for a suitable choice of S, it gives rise to drawings with worst-case optimal angular resolution. In addition, we prove that every such set S can be used to construct 2-bend upward planar drawings of n-vertex planar st-graphs with at most 4n − 9 bends in total. Our main tool is a constructive technique that runs in linear time.

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Extending Upward Planar Graph Drawings

Lozzo

Battista

Frati

2019

Lecture Notes in Computer Science

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In this paper we study the computational complexity of the Upward Planarity Extension problem, which takes in input an upward planar drawing ΓH of a subgraph H of a directed graph G and asks whether ΓH can be extended to an upward planar drawing of G. Our study fits into the line of research on the extensibility of partial representations, which has recently become a mainstream in Graph Drawing. We show the following results.-First, we prove that the Upward Planarity Extension problem is NP-complete, even if G has a prescribed upward embedding, the vertex set of H coincides with the one of G, and H contains no edge. -Second, we show that the Upward Planarity Extension problem can be solved in O(n log n) time if G is an n-vertex upward planar st-graph. This result improves upon a known O(n 2 )-time algorithm, which however applies to all n-vertex single-source upward planar graphs. -Finally, we show how to solve in polynomial time a surprisingly difficult version of the Upward Planarity Extension problem, in which G is a directed path or cycle with a prescribed upward embedding, H contains no edges, and no two vertices share the same y-coordinate in ΓH .that the drawing of G we seek has to respect a given upward embedding, i.e., a left-to-right order of the edges entering and exiting each vertex. Related problems. Level planar drawings are special upward planar drawings. Klemz and Rote studied the Ordered Level Planarity (OLP) problem [26], where a partial drawing of a level graph is given containing all the vertices and no edges. The problem asks for the existence of a level planar drawing of the graph extending the partial one. They show a tight border of tractability for the problem by proving NP-completeness even if no three vertices have the same y-coordinate and by providing a linear-time algorithm if no two vertices have the same y-coordinate. Brückner and Rutter studied the Partial Level Planarity (PLP) problem [9], that is, the extensibility of a partial drawing of a level graph, which might contain (not necessarily all) vertices and edges. Beside proving NP-completeness even for connected graphs, they provided a quadratic-time algorithm for single-source graphs.The NP-hardness of the Upward Planarity Testing problem [19] directly implies the NP-hardness of the UPE problem, as the former coincides with the special case of the latter in which the partial graph is the empty graph. Further, we have that any instance of the OLP problem in which no λ vertices have the same y-coordinate can be transformed in linear time into an equivalent instance of the UPE problem in which the partial graph contains all the vertices and no edges, and no λ vertices have the same y-coordinate in the partial drawing; moreover, a linear-time reduction can also be performed in the opposite direction. As a consequence of these reductions and of the cited results about the complexity of the OLP problem, we obtain that the UPE problem is NP-hard even if the partial graph contains all the vertices and no edges, and no three vertices s...

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Planar L-Drawings of Directed Graphs

Cited by 14 publications

References 17 publications

Planar L-Drawings of Bimodal Graphs

Planar L-Drawings of Bimodal Graphs

Universal Slope Sets for Upward Planar Drawings

Extending Upward Planar Graph Drawings

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