Algorithms and Bounds for L-Drawings of Directed Graphs

Angelini, Patrizio; Lozzo, Giordano Da; Bartolomeo, Marco Di; Donato, Valentino Di; Patrignani, Maurizio; Roselli, Vincenzo; Tollis, Ioannis G.

doi:10.1142/s0129054118410010

Cited by 8 publications

(7 citation statements)

References 11 publications

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“…They are called rook-drawings when the edges are straight lines 58,59 and L-drawings when the edges have an orthogonal L-shape, that is, when each edge consists of exactly a vertical segment and a horizontal segment. 11 Our 2D drawing model will be called cc-L-drawing , because it is an L-drawing with rectangular regions that enclose the different connected components. Note that, since no vertices share the same x -coordinate or the same y -coordinate, an L-shaped edge in the drawing can always be added without intersecting any vertex, except its end-vertices.…”

Section: Models Metrics and Algorithmsmentioning

confidence: 99%

“…We describe two different visualization models and related online algorithms, one for the 1-dimensional space and the other for the 2-dimensional space. The 1-dimensional visualization model adopts an arc diagram style, 10 while the 2-dimensional model exploits a variant of orthogonal drawings , 11,12 that is, drawings whose vertices are mapped to points of an integer grid and edges are drawn as chains of horizontal and vertical segments. In both models the connected components are regarded as clusters and are drawn inside rectangular regions.…”

Section: Introductionmentioning

confidence: 99%

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Stable visualization of connected components in dynamic graphs

Giacomo

Didimo

Kaufmann

et al. 2020

Information Visualization

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One of the primary goals of many systems for the visual analysis of dynamically changing networks is to maintain the stability of the drawing throughout the sequence of graph changes. We investigate the scenario where the changes are determined by a stream of events, each being either an edge addition or an edge removal. The visualization must be updated immediately after each new event is received. Our main goal is to provide the user with an intuitive visualization that highlights the different connected components of the graph while preserving the user’s mental map after each event. The drawing stability is measured in terms of changes in the orthogonal relationships between vertices of two consecutive drawings. We describe two different visualization models, one for the 1-dimensional space and the other for the 2-dimensional space. In both models the connected components are drawn inside rectangular regions. To validate our approach, we report the results of an experimental analysis that compares the drawing stability of the online algorithm with that of an offline algorithm that knows in advance the whole sequence of events. We also present a case study of our online algorithm on a collaboration network.

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Section: Models Metrics and Algorithmsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Stable visualization of connected components in dynamic graphs

Giacomo

Didimo

Kaufmann

et al. 2020

Information Visualization

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“…Assume first that T has a source s, a sink t and an in/out-vertex w. There are three cases: (1) The angles in T at s and t, respectively, are both 0 • (Fig. 18), ( 2) the angle at either the source or the sink -say the source -is 0 • (Fig.…”

Section: E Planar 3-treesmentioning

confidence: 99%

“…1(c)). 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.…”

Section: Introductionmentioning

confidence: 99%

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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“…4-modal embeddings, where the outgoing and the incoming edges at each vertex form up to four disjoint sequences with alternating orientations, arise in the context of planar L-drawings of digraphs. In an L-drawing of an n-vertex digraph, introduced by Angelini et al [1], vertices are placed on the n × n grid so that each vertex is assigned a unique x-coordinate and a unique y-coordinate and each edge uv (directed from u to v) is represented as a 1-bend orthogonal polyline composed of a vertical segment incident to u and of a horizontal segment incident to v. Recently, Chaplick et al [13] addressed the question of deciding the existence of planar L-drawings, i.e., L-drawings whose edges might possibly overlap but do not cross and observe that the existence of a 4-modal embedding is a necessary condition for a digraph to admit such a representation (Fig. 1a).…”

Section: Introductionmentioning

confidence: 99%

Computing k-Modal Embeddings of Planar Digraphs

Besa¹,

Lozzo²,

Goodrich³

2019

Preprint

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Given a planar digraph G and a positive even integer k, an embedding of G in the plane is k-modal, if every vertex of G is incident to at most k pairs of consecutive edges with opposite orientations, i.e., the incoming and the outgoing edges at each vertex are grouped by the embedding into at most k sets of consecutive edges with the same orientation. In this paper, we study the k-Modality problem, which asks for the existence of a k-modal embedding of a planar digraph. This combinatorial problem is at the very core of a variety of constrained embedding questions for planar digraphs and flat clustered networks.First, since the 2-Modality problem can be easily solved in linear time, we consider the general k-Modality problem for any value of k > 2 and show that the problem is NP-complete for planar digraphs of maximum degree ∆ ≥ k + 3. We relate its computational complexity to that of two notions of planarity for flat clustered networks: Planar Intersection-Link and Planar NodeTrix representations. This allows us to answer in the strongest possible way an open question by Di Giacomo et al. [GD17], concerning the complexity of constructing planar NodeTrix representations of flat clustered networks with small clusters, and to address a research question by Angelini et al. [JGAA17], concerning intersection-link representations based on geometric objects that determine complex arrangements. On the positive side, we provide a simple FPT algorithm for partial 2-trees of arbitrary degree, whose running time is exponential in k and linear in the input size.Second, motivated by the recently-introduced planar L-drawings of planar digraphs [GD17], which require the computation of a 4-modal embedding, we focus our attention on k = 4. On the algorithmic side, we show a complexity dichotomy for the 4-Modality problem with respect to ∆, by providing a linear-time algorithm for planar digraphs with ∆ ≤ 6. This algorithmic result is based on decomposing the input digraph into its blocks via BC-trees and each of these blocks into its triconnected components via SPQR-trees. In particular, we are able to show that the constraints imposed on the embedding by the rigid triconnected components can be tackled by means of a small set of reduction rules and discover that the algorithmic core of the problem lies in special instances of NAESAT, which we prove to be always NAE-satisfiable-a result of independent interest that improves on Porschen et al.Finally, on the combinatorial side, we consider outerplanar digraphs and show that any such a digraph always admits a k-modal embedding with k = 4 and that this value of k is best possible for the digraphs in this family.

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Algorithms and Bounds for L-Drawings of Directed Graphs

Cited by 8 publications

References 11 publications

Stable visualization of connected components in dynamic graphs

Stable visualization of connected components in dynamic graphs

Planar L-Drawings of Bimodal Graphs

Computing k-Modal Embeddings of Planar Digraphs

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