Classification of Planar Upward Embedding

Auer, Christopher; Bachmaier, Christian; Brandenburg, Franz J.; Gleißner, Andreas

doi:10.1007/978-3-642-25878-7_39

Cited by 5 publications

(8 citation statements)

References 19 publications

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“…The notion of level planarity in this setting goes by the name of Radial Level Planarity and is known to be decidable in linear time [3]. This setting is equivalent to the one in which the level graph is embedded on the "standing cylinder": Here, the vertices have to be placed on the circles defined by the intersection of the cylinder surface S with planes parallel to the cylinder bases, and the edges are curves on S monotone with respect to the cylinder axis; see [2,3,9] and Fig. 1(b).…”

Section: Introduction and Overviewmentioning

confidence: 99%

Beyond Level Planarity

Angelini

Lozzo

Battista

et al. 2016

Lecture Notes in Computer Science

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In this paper we settle the computational complexity of two open problems related to the extension of the notion of level planarity to surfaces different from the plane. Namely, we show that the problems of testing the existence of a level embedding of a level graph on the surface of the rolling cylinder or on the surface of the torus, respectively known by the name of Cyclic Level Planarity and Torus Level Planarity, are polynomial-time solvable. Moreover, we show a complexity dichotomy for testing the Simultaneous Level Planarity of a set of level graphs, with respect to both the number of level graphs and the number of levels.

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Section: Introduction and Overviewmentioning

confidence: 99%

Beyond Level Planarity

Angelini

Lozzo

Battista

et al. 2016

Lecture Notes in Computer Science

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“…Note that the cylinder (and the sphere) and the plane are equivalent with respect to planarity, i. e., every planar graph is cylindric planar and vice versa. This does no longer hold true for planar upward drawings of directed planar graphs [4,10], and for linear layouts, as we shall show in Section 3.1.…”

Section: Introductionmentioning

confidence: 94%

“…The head and tail correspond to the regions above and below the front line, respectively. If e = (v, w) is an edge with e ∈ E ⊥ (v), e. g., edge (2,4) in Fig. 2(c) in the LC layout, then e ∈ E h (v) in the deque layout.…”

Section: Theorem 1 a Graph Is A Deque Graph If And Only If It Is Lc Pmentioning

confidence: 99%

“…The content of the deque is given by a vertical scan line from bottom to top. The dashed edge (3,8) causes an error at vertex 7, since the content is ((6, 7), (6,8), (5,8), (4,7), (3,8), (3,7), (5, 7)) and (3,8) blocks the removal of (4, 7). Furthermore, at each vertex the incoming edges from predecessors must be removed before the outgoing edges to successors.…”

Section: Theorem 1 a Graph Is A Deque Graph If And Only If It Is Lc Pmentioning

confidence: 99%

“…1(b). The dashed edge and (1,4) nest and invalidate the layout. The LIFO principle of a stack is visualized by a rainbow of nesting edges.…”

Section: Introductionmentioning

confidence: 99%

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Data Structures and their Planar Graph Layouts

Auer¹,

Bachmaier²,

Brandenburg³

et al. 2018

JGAA

Self Cite

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In a stack layout, also known as book embedding, the vertices of a graph are placed on a line and an edge is a data item that is pushed on the stack at the left vertex and removed at the right vertex. The LIFO principle of the stack is represented by a rainbow of nesting edges.We introduce linear cylindric drawings for the representation of the working principles of fundamental data structures including stack, queue, double-stack and deque. The resulting graphs are called stack (queue, double-stack, deque) graphs. We characterize the feasibility of a sequence of insertions and removals by planarity and use the graph classes to compare the relative power of these data structures.In particular, we show that the deque graphs are the linear cylindric planar graphs, and are the subgraphs of the planar graphs with a Hamiltonian path. In comparison, the double-stack graphs are the graphs with a linear layout in the plane, and are known as the subgraphs of the planar graphs with a Hamiltonian cycle. Hence, the power of the queue mode of a deque is expressed both by the differences between Hamiltonian path and Hamiltonian cycle and by linear layouts on the cylinder and in the plane. It is also reflected in the dual graph. Linear cylindric drawings provide an intuitive planar representation of the FIFO principle of a queue. We show that a queue graph augmented by a Hamiltonian path has a dual of the same type, and that the dual has an Eulerian path. Finally, we study recognition problems.

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Quasi-Upward Planar Drawings of Mixed Graphs with Few Bends: Heuristics and Exact Methods

Binucci

Didimo

2014

Algorithms and Computation

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Classification of Planar Upward Embedding

Cited by 5 publications

References 19 publications

Beyond Level Planarity

Beyond Level Planarity

Data Structures and their Planar Graph Layouts

Quasi-Upward Planar Drawings of Mixed Graphs with Few Bends: Heuristics and Exact Methods

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