Geometric Thickness of Complete Graphs

Dillencourt, Michael B.; Eppstein, David; Hirschberg, Daniel S.

doi:10.1007/3-540-37623-2_8

Cited by 21 publications

(37 citation statements)

References 16 publications

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“…By Theorem 1.4 there is an infinite family of bipartite expanders with simple thickness 2. Furthermore, one may subdivide each edge twice in the above construction, and then draw each edge straight to obtain an infinite family of bipartite expanders with geometric thickness 2 (see [3,10,18]). …”

Section: Thicknessmentioning

confidence: 99%

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Dujmović¹,

Sidiropoulos²,

Wood³

2016

Chicago J. of Theoretical Comp. Sci.

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Section: Thicknessmentioning

confidence: 99%

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Dujmović¹,

Sidiropoulos²,

Wood³

2016

Chicago J. of Theoretical Comp. Sci.

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“…It is unknown whether for some constant ε > 0, every geometric drawing of K n has thickness at most (1 − ε)n; see [17]. Dillencourt et al [26] studied the geometric thickness of K n , and proved that 5…”

Section: Open Problemsmentioning

confidence: 99%

“…Testing whether a graph has book thickness at most 2 is N P-complete [76]. Dillencourt et al [26] asked what is the complexity of determining the geometric thickness of a given graph? The same question can be asked for all of the other parameters discussed in this paper.…”

Section: Open Problemsmentioning

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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“…Modified force-directed methods are used to visualize general graphs simultaneously such that the mental map is preserved up in [7]. Conceptually, the problem of simultaneously embedding graphs is the reverse of the geometric thickness problem [5].…”

Section: Related Workmentioning

confidence: 99%

An Interactive Multi-user System for Simultaneous Graph Drawing

Kobourov

Pitta

2005

Graph Drawing

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Abstract. In this paper we consider the problem of simultaneous drawing of two graphs. The goal is to produce aesthetically pleasing drawings for the two graphs by means of a heuristic algorithm and with human assistance. Our implementation uses the DiamondTouch table, a multiuser, touch-sensitive input device, to take advantage of direct physical interaction of several users working collaboratively. The system can be downloaded at http://dt.cs.arizona.edu where it is also available as an applet.

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Geometric Thickness of Complete Graphs

Cited by 21 publications

References 16 publications

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Graph Treewidth and Geometric Thickness Parameters

An Interactive Multi-user System for Simultaneous Graph Drawing

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