Upward drawings of triconnected digraphs

Bertolazzi, Paola; Battista, Giuseppe Di; Liotta, Giuseppe; Mannino, Carlo

doi:10.1007/bf01188716

Cited by 118 publications

(170 citation statements)

References 23 publications

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“…This question is motivated by the existence of many Graph Drawing problems on planar graphs that are in general NP-hard and that become polynomial-time solvable if the embedding is fixed. Testing if a graph admits an orthogonal planar drawing with at most k bends [15,9] or if a graph admits an upward planar drawing [1,9] are examples of such problems.…”

Section: Introductionmentioning

confidence: 99%

Efficient C-Planarity Testing for Embedded Flat Clustered Graphs with Small Faces

Battista¹,

Frati²

2009

JGAA

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Let C be a clustered graph and suppose that the planar embedding of its underlying graph is fixed. Is testing the c-planarity of C easier than in the variable embedding setting? In this paper we give a first contribution towards answering the above question. Namely, we characterize c-planar embedded flat clustered graphs with at most five vertices per face and give an efficient testing algorithm for such graphs. The results are based on a more general methodology that sheds new light on the c-planarity testing problem.

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Section: Introductionmentioning

confidence: 99%

Efficient C-Planarity Testing for Embedded Flat Clustered Graphs with Small Faces

Battista¹,

Frati²

2009

JGAA

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“…We remark that the approach of [1] via a variant of upward embeddings for directed graphs in our settings has several problems that seem quite hard to overcome, the main one being the fact that the result of Bertolazzi et al [4] does not extend, at least not in a natural way, to the drawings on the rolling cylinder, see e.g., Auer et al [3] for the definition of these drawings. We are not aware of a polynomial-time algorithm for the corresponding problem, nor a corresponding NP-hardness result, and find the corresponding algorithmic question interesting and related to our problem.…”

Section: Preliminariesmentioning

confidence: 99%

C-Planarity of Embedded Cyclic c-Graphs

Fulek

2016

Lecture Notes in Computer Science

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Abstract. We show that c-planarity is solvable in quadratic time for flat clustered graphs with three clusters if the combinatorial embedding of the underlying graph is fixed. In simpler graph-theoretical terms our result can be viewed as follows. Given a graph G with the vertex set partitioned into three parts embedded on a 2-sphere, our algorithm decides if we can augment G by adding edges without creating an edge-crossing so that in the resulting spherical graph the vertices of each part induce a connected sub-graph. We proceed by a reduction to the problem of testing the existence of a perfect matching in planar bipartite graphs. We formulate our result in a slightly more general setting of cyclic clustered graphs, i.e., the simple graph obtained by contracting each cluster, where we disregard loops and multi-edges, is a cycle.

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“…There are, however, several results for special problem variants or graph classes: Upward planarity testing is polynomial when the embedding of the graph (cyclic order of the edges around their incident vertices) is prespecified [2]. Furthermore, bipartite DAGs are upward planar if and only if they are planar [10] and there are polynomial algorithms for outerplanar [18] and series-parallel graphs [11].…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore, bipartite DAGs are upward planar if and only if they are planar [10] and there are polynomial algorithms for outerplanar [18] and series-parallel graphs [11]. In [17], a quadratic testing algorithm-later improved to linear time [2]-has been proposed for single-source DAGs, i.e., graphs with a unique vertex without any ingoing edges. Several FPT algorithms were proposed, e.g., by using the number of triconnected components t as their central fixed parameter.…”

Section: Introductionmentioning

confidence: 99%

Upward Planarity Testing via SAT

Chimani

Zeranski

2013

Graph Drawing

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Abstract.A directed acyclic graph is upward planar if it allows a drawing without edge crossings where all edges are drawn as curves with monotonously increasing y-coordinates. The problem to decide whether a graph is upward planar or not is NP-complete in general, and while special graph classes are polynomial time solvable, there is not much known about solving the problem for general graphs in practice. The only attempt so far was a branch-and-bound algorithm over the graph's triconnectivity structure which was able to solve sparse graphs.In this paper, we propose a fundamentally different approach, based on the seemingly novel concept of ordered embeddings. We carefully model the problem as a special SAT instance, i.e., a logic formula for which we check satisfiability. Solving these SAT instances allows us to decide upward planarity for arbitrary graphs. We then show experimentally that this approach seems to dominate the known alternative approaches and is able to solve traditionally used graph drawing benchmarks effectively.

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Upward drawings of triconnected digraphs

Cited by 118 publications

References 23 publications

Efficient C-Planarity Testing for Embedded Flat Clustered Graphs with Small Faces

Efficient C-Planarity Testing for Embedded Flat Clustered Graphs with Small Faces

C-Planarity of Embedded Cyclic c-Graphs

Upward Planarity Testing via SAT

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