1,183 research outputs found
Convexity-Increasing Morphs of Planar Graphs
We study the problem of convexifying drawings of planar graphs. Given any
planar straight-line drawing of an internally 3-connected graph, we show how to
morph the drawing to one with strictly convex faces while maintaining planarity
at all times. Our morph is convexity-increasing, meaning that once an angle is
convex, it remains convex. We give an efficient algorithm that constructs such
a morph as a composition of a linear number of steps where each step either
moves vertices along horizontal lines or moves vertices along vertical lines.
Moreover, we show that a linear number of steps is worst-case optimal.
To obtain our result, we use a well-known technique by Hong and Nagamochi for
finding redrawings with convex faces while preserving y-coordinates. Using a
variant of Tutte's graph drawing algorithm, we obtain a new proof of Hong and
Nagamochi's result which comes with a better running time. This is of
independent interest, as Hong and Nagamochi's technique serves as a building
block in existing morphing algorithms.Comment: Preliminary version in Proc. WG 201
Linear transformation distance for bichromatic matchings
Let be a set of points in general position, where is a
set of blue points and a set of red points. A \emph{-matching}
is a plane geometric perfect matching on such that each edge has one red
endpoint and one blue endpoint. Two -matchings are compatible if their
union is also plane.
The \emph{transformation graph of -matchings} contains one node for each
-matching and an edge joining two such nodes if and only if the
corresponding two -matchings are compatible. In SoCG 2013 it has been shown
by Aloupis, Barba, Langerman, and Souvaine that this transformation graph is
always connected, but its diameter remained an open question. In this paper we
provide an alternative proof for the connectivity of the transformation graph
and prove an upper bound of for its diameter, which is asymptotically
tight
Facets of Planar Graph Drawing
This thesis makes a contribution to the field of Graph Drawing, with a focus on the planarity drawing convention. The following three problems are considered.
(1) Ordered Level Planarity:
We introduce and study the problem Ordered Level Planarity which asks for a planar drawing of a graph such that vertices are placed at prescribed positions in the plane and such that every edge is realized as a y-monotone curve. This can be interpreted as a variant of Level Planarity in which the vertices on each level appear in a prescribed total order. We establish a complexity dichotomy with respect to both the maximum degree and the level-width, that is, the maximum number of vertices that share a level. Our study of Ordered Level Planarity is motivated by connections to several other graph drawing problems. With reductions from Ordered Level Planarity, we show NP-hardness of multiple problems whose complexity was previously open, and strengthen several previous hardness results. In particular, our reduction to Clustered Level Planarity generates instances with only two nontrivial clusters. This answers a question posed by Angelini, Da Lozzo, Di Battista, Frati, and Roselli [2015]. We settle the complexity of the Bi-Monotonicity problem, which was proposed by Fulek, Pelsmajer, Schaefer, and Stefankovic [2013]. We also present a reduction to Manhattan Geodesic Planarity, showing that a previously [2009] claimed polynomial time algorithm is incorrect unless P=NP.
(2) Two-page book embeddings of triconnected planar graphs:
We show that every triconnected planar graph of maximum degree five is a subgraph of a Hamiltonian planar graph or, equivalently, it admits a two-page book embedding. In fact, our result is more general: we only require vertices of separating 3-cycles to have degree at most five, all other vertices may have arbitrary degree. This degree bound is tight: we describe a family of triconnected planar graphs that cannot be realized on two pages and where every vertex of a separating 3-cycle has degree at most six. Our results strengthen earlier work by Heath [1995] and by Bauernöppel [1987] and, independently, Bekos, Gronemann, and Raftopoulou [2016], who showed that planar graphs of maximum degree three and four, respectively, can always be realized on two pages. The proof is constructive and yields a quadratic time algorithm to realize the given graph on two pages.
(3) Convexity-increasing morphs:
We study the problem of convexifying drawings of planar graphs. Given any planar straight-line drawing of an internally 3-connected graph, we show how to morph the drawing to one with strictly convex faces while maintaining planarity at all times. Our morph is convexity-increasing, meaning that once an angle is convex, it remains convex. We give an efficient algorithm that constructs such a morph as a composition of a linear number of steps where each step either moves vertices along horizontal lines or moves vertices along vertical lines. Moreover, we show that a linear number of steps is worst-case optimal.Diese Arbeit behandelt drei unterschiedliche Problemstellungen aus der Disziplin des Graphenzeichnens (Graph Drawing). Bei jedem der behandelten Probleme ist die gesuchte Darstellung planar.
(1) Ordered Level Planarity:
Wir fĂŒhren das Problem Ordered Level Planarity ein, bei dem es darum geht, einen Graph so zu zeichnen, dass jeder Knoten an einer vorgegebenen Position der Ebene platziert wird und die Kanten als y-monotone Kurven dargestellt werden. Dies kann als eine Variante von Level Planarity interpretiert werden, bei der die Knoten jedes Levels in einer vorgeschriebenen Reihenfolge platziert werden mĂŒssen. Wir klassifizieren die Eingaben bezĂŒglich ihrer KomplexitĂ€t in AbhĂ€ngigkeit von sowohl dem Maximalgrad, als auch der maximalen Anzahl von Knoten, die demselben Level zugeordnet sind. Wir motivieren die Ergebnisse, indem wir Verbindungen zu einigen anderen Graph Drawing Problemen herleiten: Mittels Reduktionen von Ordered Level Planarity zeigen wir die NP-Schwere einiger Probleme, deren KomplexitĂ€t bislang offen war. Insbesondere wird gezeigt, dass Clustered Level Planarity bereits fĂŒr Instanzen mit zwei nichttrivialen Clustern NP-schwer ist, was eine Frage von Angelini, Da Lozzo, Di Battista, Frati und Roselli [2015] beantwortet. Wir zeigen die NP-Schwere des Bi-Monotonicity Problems und beantworten damit eine Frage von Fulek, Pelsmajer, Schaefer und Stefankovic [2013]. AuĂerdem wird eine Reduktion zu Manhattan Geodesic Planarity angegeben. Dies zeigt, dass ein bestehender [2009] Polynomialzeitalgorithmus fĂŒr dieses Problem inkorrekt ist, es sei denn, dass P=NP ist.
(2) Bucheinbettungen von dreifach zusammenhÀngenden planaren Graphen mit zwei Seiten:
Wir zeigen, dass jeder dreifach zusammenhĂ€ngende planare Graph mit Maximalgrad 5 Teilgraph eines Hamiltonischen planaren Graphen ist. Dies ist Ă€quivalent dazu, dass ein solcher Graph eine Bucheinbettung auf zwei Seiten hat. Der Beweis ist konstruktiv und zeigt in der Tat sogar, dass es fĂŒr die Realisierbarkeit nur notwendig ist, den Grad von Knoten separierender 3-Kreise zu beschrĂ€nken - die ĂŒbrigen Knoten können beliebig hohe Grade aufweisen. Dieses Ergebnis ist bestmöglich: Wenn die Gradschranke auf 6 abgeschwĂ€cht wird, gibt es Gegenbeispiele. Diese Ergebnisse verbessern Resultate von Heath [1995] und von Bauernöppel [1987] und, unabhĂ€ngig davon, Bekos, Gronemann und Raftopoulou [2016], die gezeigt haben, dass planare Graphen mit Maximalgrad 3 beziehungsweise 4 auf zwei Seiten realisiert werden können.
(3) KonvexitÀtssteigernde Deformationen:
Wir zeigen, dass jede planare geradlinige Zeichnung eines intern dreifach zusammenhÀngenden planaren Graphen stetig zu einer solchen deformiert werden kann, in der jede FlÀche ein konvexes Polygon ist. Dabei erhÀlt die Deformation die PlanaritÀt und ist konvexitÀtssteigernd - sobald ein Winkel konvex ist, bleibt er konvex. Wir geben einen effizienten Algorithmus an, der eine solche Deformation berechnet, die aus einer asymptotisch optimalen Anzahl von Schritten besteht. In jedem Schritt bewegen sich entweder alle Knoten entlang horizontaler oder entlang vertikaler Geraden
Approximation algorithms for node-weighted prize-collecting Steiner tree problems on planar graphs
We study the prize-collecting version of the Node-weighted Steiner Tree
problem (NWPCST) restricted to planar graphs. We give a new primal-dual
Lagrangian-multiplier-preserving (LMP) 3-approximation algorithm for planar
NWPCST. We then show a ()-approximation which establishes a
new best approximation guarantee for planar NWPCST. This is done by combining
our LMP algorithm with a threshold rounding technique and utilizing the
2.4-approximation of Berman and Yaroslavtsev for the version without penalties.
We also give a primal-dual 4-approximation algorithm for the more general
forest version using techniques introduced by Hajiaghay and Jain
A survey of parameterized algorithms and the complexity of edge modification
The survey is a comprehensive overview of the developing area of parameterized algorithms for graph modification problems. It describes state of the art in kernelization, subexponential algorithms, and parameterized complexity of graph modification. The main focus is on edge modification problems, where the task is to change some adjacencies in a graph to satisfy some required properties. To facilitate further research, we list many open problems in the area.publishedVersio
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