
Planar graphs as Lintersection or Lcontact graphs
The Lintersection graphs are the graphs that have a representation as i...
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Finding a Maximum Minimal Separator: Graph Classes and FixedParameter Tractability
We study the problem of finding a maximum cardinality minimal separator ...
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Recognizing Generalized Transmission Graphs of Line Segments and Circular Sectors
Suppose we have an arrangement A of n geometric objects x_1, ..., x_n ⊆R...
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Rigid Foldability is NPHard
In this paper, we show that deciding rigid foldability of a given crease...
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Approximating Dominating Set on Intersection Graphs of Lframes
We consider the Dominating Set (DS) problem on the intersection graphs o...
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Axesparallel unit disk graph recognition is NPhard
Unit disk graphs are the intersection graphs of unit diameter disks in t...
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Linking disjoint segments into a simple polygon is hard
Deciding whether a family of disjoint line segments in the plane can be ...
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Finding Geometric Representations of Apex Graphs is NPHard
Planar graphs can be represented as intersection graphs of different types of geometric objects in the plane, e.g., circles (Koebe, 1936), line segments (Chalopin & Gonçalves, 2009), Lshapes (Gonçalves et al, 2018). For general graphs, however, even deciding whether such representations exist is often NPhard. We consider apex graphs, i.e., graphs that can be made planar by removing one vertex from them. We show, somewhat surprisingly, that deciding whether geometric representations exist for apex graphs is NPhard. More precisely, we show that for every positive integer k, recognizing every graph class 𝒢 which satisfies PURE2DIR⊆𝒢⊆1STRING is NPhard, even when the input graphs are apex graphs of girth at least k. Here, PURE2DIR is the class of intersection graphs of axisparallel line segments (where intersections are allowed only between horizontal and vertical segments) and 1STRING is the class of intersection graphs of simple curves (where two curves share at most one point) in the plane. This partially answers an open question raised by Kratochvíl & Pergel (2007). Most known NPhardness reductions for these problems are from variants of 3SAT. We reduce from the PLANAR HAMILTONIAN PATH COMPLETION problem, which uses the more intuitive notion of planarity. As a result, our proof is much simpler and encapsulates several classes of geometric graphs.
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