Book Embeddability of Series–Parallel Digraphs

Giacomo, Emilio Di; Didimo, Walter; Liotta, Giuseppe; Wismath, Stephen K.

doi:10.1007/s00453-005-1185-7

Cited by 37 publications

(31 citation statements)

References 23 publications

(81 reference statements)

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“…At most one bend per edge is sufficient if and only if the graph has a 2-page upward book embedding [19]. Our result: At most two bends per edge are sufficient (Theorem 3).…”

Section: Planar Graphs Upward Planar Digraphsmentioning

confidence: 78%

“…In [19] it was proven that an upward planar digraph has a 1-bend UPSE if and only if it has a 2-page upward book embedding (this will be formally stated in Theorem 4 of Section 4.2). It is also known that the following classes of digraphs admit a 2-page upward book embedding: tree dags [33], unicyclic dags [33], and two-terminal series-parallel digraphs [19].…”

Section: Upward Point-set Embeddability Without a Given Mappingmentioning

confidence: 99%

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Computing upward topological book embeddings of upward planar digraphs

Giordano

Liotta

Mchedlidze

et al. 2015

Journal of Discrete Algorithms

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We describe a unified approach for studying book, point-set, and simultaneous embeddability problems of upward planar digraphs. The approach is based on a linear time strategy to compute an upward planar drawing of an upward planar digraph such that all vertices are collinear. Besides having impact in relevant application domains of graph drawing and computational geometry, the presented results open new research directions in the area of upward planarity with constraints of the positions of the vertices.

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“…At most one bend per edge is sufficient if and only if the graph has a 2-page upward book embedding [19]. Our result: At most two bends per edge are sufficient (Theorem 3).…”

Section: Planar Graphs Upward Planar Digraphsmentioning

confidence: 78%

Section: Upward Point-set Embeddability Without a Given Mappingmentioning

confidence: 99%

Computing upward topological book embeddings of upward planar digraphs

Giordano

Liotta

Mchedlidze

et al. 2015

Journal of Discrete Algorithms

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“…Rengarajan and Madhavan [67] proved that every seriesparallel graph has a 2-page book embedding (see also [24]); that is, bt(T 2 ) ≤ 2. Note that bt(T 2 ) = 2 since there are series-parallel graphs that are not outerplanar, K 2,3 being the primary example.…”

Section: Theorem 3 the Maximum Book Thickness And Maximum Book Arbormentioning

confidence: 99%

Graph Treewidth and Geometric Thickness Parameters

Dujmović

Wood

2007

Discrete Comput Geom

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Consider a drawing of a graph $G$ in the plane such that crossing edges are coloured differently. The minimum number of colours, taken over all drawings of $G$, is the classical graph parameter "thickness". By restricting the edges to be straight, we obtain the "geometric thickness". By further restricting the vertices to be in convex position, we obtain the "book thickness". This paper studies the relationship between these parameters and treewidth. Our first main result states that for graphs of treewidth $k$, the maximum thickness and the maximum geometric thickness both equal $\lceil{k/2}\rceil$. This says that the lower bound for thickness can be matched by an upper bound, even in the more restrictive geometric setting. Our second main result states that for graphs of treewidth $k$, the maximum book thickness equals $k$ if $k \leq 2$ and equals $k+1$ if $k \geq 3$. This refutes a conjecture of Ganley and Heath [Discrete Appl. Math. 109(3):215-221, 2001]. Analogous results are proved for outerthickness, arboricity, and star-arboricity.Comment: A preliminary version of this paper appeared in the "Proceedings of the 13th International Symposium on Graph Drawing" (GD '05), Lecture Notes in Computer Science 3843:129-140, Springer, 2006. The full version was published in Discrete & Computational Geometry 37(4):641-670, 2007. That version contained a false conjecture, which is corrected on page 26 of this versio

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“…Rengarajan and Veni Madhavan [67] proved that every seriesparallel graph has a 2-page book embedding (also see [24]); that is, bt(T 2 ) ≤ 2. Note that bt(T 2 ) = 2 since there are series-parallel graphs that are not outerplanar, K 2,3 being the primary example.…”

Section: Theorem 3 the Maximum Book Thickness And Maximum Book Arbormentioning

confidence: 99%

Graph Treewidth and Geometric Thickness Parameters

Dujmović

Wood

2006

Graph Drawing

View full text Add to dashboard Cite

Consider a drawing of a graph G in the plane such that crossing edges are coloured differently. The minimum number of colours, taken over all drawings of G, is the classical graph parameter thickness. By restricting the edges to be straight, we obtain the geometric thickness. By additionally restricting the vertices to be in convex position, we obtain the book thickness. This paper studies the relationship between these parameters and treewidth.Our first main result states that for graphs of treewidth k, the maximum thickness and the maximum geometric thickness both equal ⌈k/2⌉. This says that the lower bound for thickness can be matched by an upper bound, even in the more restrictive geometric setting. Our second main result states that for graphs of treewidth k, the maximum book thickness equals k if k ≤ 2 and equals k + 1 if k ≥ 3. This refutes a conjecture of Ganley and Heath [Discrete Appl. Math. 109(3):215-221, 2001]. Analogous results are proved for outerthickness, arboricity, and star-arboricity.

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Book Embeddability of Series–Parallel Digraphs

Cited by 37 publications

References 23 publications

Computing upward topological book embeddings of upward planar digraphs

Computing upward topological book embeddings of upward planar digraphs

Graph Treewidth and Geometric Thickness Parameters

Graph Treewidth and Geometric Thickness Parameters

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