4 research outputs found
Recognizing Geometric Intersection Graphs Stabbed by a Line
In this paper, we determine the computational complexity of recognizing two
graph classes, \emph{grounded L}-graphs and \emph{stabbable grid intersection}
graphs. An L-shape is made by joining the bottom end-point of a vertical
() segment to the left end-point of a horizontal () segment. The top
end-point of the vertical segment is known as the {\em anchor} of the L-shape.
Grounded L-graphs are the intersection graphs of L-shapes such that all the
L-shapes' anchors lie on the same horizontal line. We show that recognizing
grounded L-graphs is NP-complete. This answers an open question asked by
Jel{\'\i}nek \& T{\"o}pfer (Electron. J. Comb., 2019).
Grid intersection graphs are the intersection graphs of axis-parallel line
segments in which two vertical (similarly, two horizontal) segments cannot
intersect. We say that a (not necessarily axis-parallel) straight line
stabs a segment , if intersects . A graph is a stabbable grid
intersection graph () if there is a grid intersection representation
of in which the same line stabs all its segments. We show that recognizing
graphs is -complete, even on a restricted class of graphs. This
answers an open question asked by Chaplick \etal (\textsc{O}rder, 2018).Comment: 18 pages, 11 Figure
Finding Geometric Representations of Apex Graphs is NP-Hard
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 \& Gon{\c{c}}alves, 2009), \textsc{L}-shapes (Gon{\c{c}}alves et al,
2018). For general graphs, however, even deciding whether such representations
exist is often -hard. 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
-hard.
More precisely, we show that for every positive integer , recognizing
every graph class which satisfies \textsc{PURE-2-DIR} \subseteq
\mathcal{G} \subseteq \textsc{1-STRING} is -hard, even when the input
graphs are apex graphs of girth at least . Here, is the class
of intersection graphs of axis-parallel line segments (where intersections are
allowed only between horizontal and vertical segments) and \textsc{1-STRING} 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
Kratochv{\'\i}l \& Pergel (2007).
Most known -hardness reductions for these problems are from variants of
3-SAT. We reduce from the \textsc{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
Planare Graphen und ihre Dualgraphen auf Zylinderoberflächen
In this thesis, we investigates plane drawings of undirected and directed graphs on cylinder surfaces. In the case of undirected graphs, the vertices are positioned on a line that is parallel to the cylinder’s axis and the edge curves must not intersect this line. We show that a plane drawing is possible if and only if the graph is a double-ended queue (deque) graph, i. e., the vertices of the graph can be processed according to a linear order and the edges correspond to items in the deque inserted and removed at their end vertices. A surprising consequence resulting from these observations is that the deque characterizes planar graphs with a Hamiltonian path. This result extends the known characterization of planar graphs with a Hamiltonian cycle by two stacks. By these insights, we also obtain a new characterization of queue graphs and their duals. We also consider the complexity of deciding whether a graph is a deque graph and prove that it is NP-complete. By introducing a split operation, we obtain the splittable deque and show that it characterizes planarity. For the proof, we devise an algorithm that uses the splittable deque to test whether a rotation system is planar. In the case of directed graphs, we study upward plane drawings where the edge curves follow the direction of the cylinder’s axis (standing upward planarity; SUP) or they wind around the axis (rolling upward planarity; RUP). We characterize RUP graphs by means of their duals and show that RUP and SUP swap their roles when considering a graph and its dual. There is a physical interpretation underlying this characterization: A SUP graph is to its RUP dual graph as electric current passing through a conductor to the magnetic field surrounding the conductor. Whereas testing whether a graph is RUP is NP-hard in general [Bra14], for directed graphs without sources and sink, we develop a linear-time recognition algorithm that is based on our dual graph characterization of RUP graphs.Die Arbeit beschäftigt sich mit planaren Zeichnungen ungerichteter und gerichteter Graphen auf Zylinderoberflächen. Im ungerichteten Fall werden Zeichnungen betrachtet, bei denen die Knoten auf einer Linie parallel zur Zylinderachse positioniert werden und die Kanten diese Linie nicht schneiden dürfen. Es kann gezeigt werden, dass eine planare Zeichnung genau dann möglich ist, wenn die Kanten des Graphen in einer double-ended queue (Deque) verarbeitet werden können. Ebenso lassen sich dadurch Queue, Stack und Doppelstack charakterisieren. Eine überraschende Konsequenz aus diesen Erkenntnissen ist, dass die Deque genau die planaren Graphen mit Hamiltonpfad charakterisiert. Dies erweitert die bereits bekannte Charakterisierung planarer Graphen mit Hamiltonkreis durch den Doppelstack. Im gerichteten Fall müssen die Kantenkurven entweder in Richtung der Zylinderachse verlaufen (SUP-Graphen) oder sich um die Achse herumbewegen (RUP-Graphen). Die Arbeit charakterisiert RUP-Graphen und zeigt, dass RUP und SUP ihre Rollen tauschen, wenn man Graph und Dualgraph betrachtet. Der SUP-Graph verhält sich dabei zum RUP-Graphen wie elektrischer Strom durch einen Leiter zum induzierten Magnetfeld. Ausgehend von dieser Charakterisierung ist es möglich einen Linearzeit-Algorithmus zu entwickeln, der entscheidet ob ein gerichteter Graph ohne Quellen und Senken ein RUP-Graph ist, während der allgemeine Fall NP-hart ist [Bra14]