The Thickness of Graphs: A Survey

Mutzel, Petra; Odenthal, Thomas; Scharbrodt, Mark

doi:10.1007/pl00007219

Cited by 90 publications

(62 citation statements)

References 34 publications

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“…A related notion is that of graph thickness [12], defined as the minimum number of planar subgraphs whose union yields the given graph. If a graph has thickness two then it can be drawn on two layers such that each layer is crossing-free and the corresponding vertices of different layers are placed in the same locations.…”

Section: Previous Workmentioning

confidence: 99%

Simultaneous Embedding of Planar Graphs with Few Bends

Erten

Kobourov

2005

Graph Drawing

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Abstract. We present an O(n) time algorithm for simultaneous embedding of pairs of planar graphs on the O(n 2 ) × O(n 2 ) grid, with at most three bends per edge, where n is the number of vertices. For the case when the input graphs are both trees, only one bend per edge is required. We also describe an O(n) time algorithm for simultaneous embedding with fixed-edges for tree-path pairs on the O(n) × O(n 2 ) grid with at most one bend per tree-edge and no bends along path edges.

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Section: Previous Workmentioning

confidence: 99%

Simultaneous Embedding of Planar Graphs with Few Bends

Erten

Kobourov

2005

Graph Drawing

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“…, G k . See [34] for a survey on thickness. A natural question arises: what is the minimum integer k for which there is an infinite family of bipartite expanders with thickness k?…”

Section: Thicknessmentioning

confidence: 99%

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Dujmović¹,

Sidiropoulos²,

Wood³

2016

Chicago J. of Theoretical Comp. Sci.

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“…The thickness of a graph G, denoted by θ(G), is the minimum number of planar subgraphs that partition G. Thickness was first defined by Tutte [73]; see the surveys [46,60]. The outerthickness of a graph G, denoted by θ o (G), is the minimum number of outerplanar subgraphs that partition G. Outerthickness was first studied by Guy [40]; also see [31,38,41,42,52,65].…”

Section: Background Graph Theorymentioning

confidence: 99%

Graph Treewidth and Geometric Thickness Parameters

Dujmović

Wood

2006

Graph Drawing

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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 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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The Thickness of Graphs: A Survey

Cited by 90 publications

References 34 publications

Simultaneous Embedding of Planar Graphs with Few Bends

Simultaneous Embedding of Planar Graphs with Few Bends

Untitled

Graph Treewidth and Geometric Thickness Parameters

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