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Given a planar graph G on n vertices, let fix (G) denote the maximum k such that any straight line drawing of G with possible edge crossings can be made crossing-free by moving at most n − k vertices to new positions. Let ¯ v(G) denote the maximum possible number of collinear vertices in a crossing-free straight line drawing of G. In view of the relation… (More)

Given a planar graph G, we consider drawings of G in the plane where edges are represented by straight line segments (which possibly intersect). Such a drawing is specified by an injective embedding π of the ver-tex set of G into the plane. Let fix (G, π) be the maximum integer k such that there exists a crossing-free redrawing π of G which keeps k vertices… (More)

Given a planar graph G, we consider drawings of G in the plane where edges are represented by straight line segments (which possibly intersect). Such a drawing is specified by an injective embedding π of the vertex set of G into the plane. Let fix (G, π) be the maximum integer k such that there exists a crossing-free redrawing π k. Given a set of points X,… (More)

We investigate the problem of drawing graphs in 2D and 3D such that their edges (or only their vertices) can be covered by few lines or planes. We insist on straight-line edges and crossing-free drawings. This problem has many connections to other challenging graph-drawing problems such as small-area or small-volume drawings, layered or track drawings, and… (More)

In their seminal work, Mustafa and Ray [25] showed that a wide class of geometric set cover (SC) problems admit a PTAS via local search, which appears to be the most powerful approach known for such problems. Their result applies if a naturally defined " exchange graph " for two feasible solutions is planar and is based on subdividing this graph via a… (More)

It is well known that any graph admits a crossing-free straight-line drawing in R 3 and that any planar graph admits the same even in R 2. For d ∈ {2, 3}, let ρ 1 d (G) denote the minimum number of lines in R d that together can accommodate all edges of a drawing of G, where ρ 1 2 (G) is defined for planar graphs. We investigate the complexity of computing… (More)

- Thomas C. van Dijk, Martin Fink, Norbert Fischer, Fabian Lipp, Peter Markfelder, Alexander Ravsky +2 others
- ArXiv
- 2016

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