15 research outputs found
Linear-Time Algorithms for Maximum-Weight Induced Matchings and Minimum Chain Covers in Convex Bipartite Graphs
A bipartite graph is convex if the vertices in can be
linearly ordered such that for each vertex , the neighbors of are
consecutive in the ordering of . An induced matching of is a
matching such that no edge of connects endpoints of two different edges of
. We show that in a convex bipartite graph with vertices and
weighted edges, an induced matching of maximum total weight can be computed in
time. An unweighted convex bipartite graph has a representation of
size that records for each vertex the first and last neighbor
in the ordering of . Given such a compact representation, we compute an
induced matching of maximum cardinality in time.
In convex bipartite graphs, maximum-cardinality induced matchings are dual to
minimum chain covers. A chain cover is a covering of the edge set by chain
subgraphs, that is, subgraphs that do not contain induced matchings of more
than one edge. Given a compact representation, we compute a representation of a
minimum chain cover in time. If no compact representation is given, the
cover can be computed in time.
All of our algorithms achieve optimal running time for the respective problem
and model. Previous algorithms considered only the unweighted case, and the
best algorithm for computing a maximum-cardinality induced matching or a
minimum chain cover in a convex bipartite graph had a running time of
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
Recognizing Weighted Disk Contact Graphs
Disk contact representations realize graphs by mapping vertices bijectively
to interior-disjoint disks in the plane such that two disks touch each other if
and only if the corresponding vertices are adjacent in the graph. Deciding
whether a vertex-weighted planar graph can be realized such that the disks'
radii coincide with the vertex weights is known to be NP-hard. In this work, we
reduce the gap between hardness and tractability by analyzing the problem for
special graph classes. We show that it remains NP-hard for outerplanar graphs
with unit weights and for stars with arbitrary weights, strengthening the
previous hardness results. On the positive side, we present constructive
linear-time recognition algorithms for caterpillars with unit weights and for
embedded stars with arbitrary weights.Comment: 24 pages, 21 figures, extended version of a paper to appear at the
International Symposium on Graph Drawing and Network Visualization (GD) 201
Linear-Time Algorithms for Maximum-Weight Induced Matchings and Minimum Chain Covers in Convex Bipartite Graphs
A bipartite graph G=(U,V,E) is convex if the vertices in V can be linearly ordered such that for each vertex uâU, the neighbors of u are consecutive in the ordering of V. An induced matching H of G is a matching for which no edge of E connects endpoints of two different edges of H. We show that in a convex bipartite graph with n vertices and m weighted edges, an induced matching of maximum total weight can be computed in O(n+m) time. An unweighted convex bipartite graph has a representation of size O(n) that records for each vertex uâU the first and last neighbor in the ordering of V. Given such a compact representation, we compute an induced matching of maximum cardinality in O(n) time. In convex bipartite graphs, maximum-cardinality induced matchings are dual to minimum chain covers. A chain cover is a covering of the edge set by chain subgraphs, that is, subgraphs that do not contain induced matchings of more than one edge. Given a compact representation, we compute a representation of a minimum chain cover in O(n) time. If no compact representation is given, the cover can be computed in O(n+m) time. All of our algorithms achieve optimal linear running time for the respective problem and model, and they improve and generalize the previous results in several ways: The best algorithms for the unweighted problem versions had a running time of O(n2) (BrandstĂ€dt et al. in Theor. Comput. Sci. 381(1â3):260â265, 2007. https://doi.org/10.1016/j.tcs.2007.04.006). The weighted case has not been considered before
Strongly Monotone Drawings of Planar Graphs
A straight-line drawing of a graph is a monotone drawing if for each pair of
vertices there is a path which is monotonically increasing in some direction,
and it is called a strongly monotone drawing if the direction of monotonicity
is given by the direction of the line segment connecting the two vertices.
We present algorithms to compute crossing-free strongly monotone drawings for
some classes of planar graphs; namely, 3-connected planar graphs, outerplanar
graphs, and 2-trees. The drawings of 3-connected planar graphs are based on
primal-dual circle packings. Our drawings of outerplanar graphs are based on a
new algorithm that constructs strongly monotone drawings of trees which are
also convex. For irreducible trees, these drawings are strictly convex
Adjacency Graphs of Polyhedral Surfaces
We study whether a given graph can be realized as an adjacency graph of the
polygonal cells of a polyhedral surface in . We show that every
graph is realizable as a polyhedral surface with arbitrary polygonal cells, and
that this is not true if we require the cells to be convex. In particular, if
the given graph contains , , or any nonplanar -tree as a
subgraph, no such realization exists. On the other hand, all planar graphs,
, and can be realized with convex cells. The same holds for
any subdivision of any graph where each edge is subdivided at least once, and,
by a result from McMullen et al. (1983), for any hypercube.
Our results have implications on the maximum density of graphs describing
polyhedral surfaces with convex cells: The realizability of hypercubes shows
that the maximum number of edges over all realizable -vertex graphs is in
. From the non-realizability of , we obtain that
any realizable -vertex graph has edges. As such, these graphs
can be considerably denser than planar graphs, but not arbitrarily dense.Comment: To appear in Proc. SoCG 202
Outside-Obstacle Representations with All Vertices on the Outer Face
An obstacle representation of a graph consists of a set of polygonal
obstacles and a drawing of as a visibility graph with respect to the
obstacles: vertices are mapped to points and edges to straight-line segments
such that each edge avoids all obstacles whereas each non-edge intersects at
least one obstacle. Obstacle representations have been investigated quite
intensely over the last few years. Here we focus on outside-obstacle
representations (OORs) that use only one obstacle in the outer face of the
drawing. It is known that every outerplanar graph admits such a representation
[Alpert, Koch, Laison; DCG 2010].
We strengthen this result by showing that every (partial) 2-tree has an OOR.
We also consider restricted versions of OORs where the vertices of the graph
lie on a convex polygon or a regular polygon. We characterize when the
complement of a tree and when a complete graph minus a simple cycle admits a
convex OOR. We construct regular OORs for all (partial) outerpaths, cactus
graphs, and grids.Comment: Appears in the Proceedings of the 30th International Symposium on
Graph Drawing and Network Visualization (GD 2022
Linear-Time Algorithms for Maximum-Weight Induced Matchings and Minimum Chain Covers in Convex Bipartite Graphs
A bipartite graph G=(U,V,E) is convex if the vertices in V can be linearly ordered such that for each vertex uâU, the neighbors of u are consecutive in the ordering of V. An induced matching H of G is a matching for which no edge of E connects endpoints of two different edges of H. We show that in a convex bipartite graph with n vertices and m weighted edges, an induced matching of maximum total weight can be computed in O(n+m) time. An unweighted convex bipartite graph has a representation of size O(n) that records for each vertex uâU the first and last neighbor in the ordering of V. Given such a compact representation, we compute an induced matching of maximum cardinality in O(n) time. In convex bipartite graphs, maximum-cardinality induced matchings are dual to minimum chain covers. A chain cover is a covering of the edge set by chain subgraphs, that is, subgraphs that do not contain induced matchings of more than one edge. Given a compact representation, we compute a representation of a minimum chain cover in O(n) time. If no compact representation is given, the cover can be computed in O(n+m) time. All of our algorithms achieve optimal linear running time for the respective problem and model, and they improve and generalize the previous results in several ways: The best algorithms for the unweighted problem versions had a running time of O(n) (BrandstĂ€dt et al. in Theor. Comput. Sci. 381(1â3):260â265, 2007. https://doi.org/10.1016/j.tcs.2007.04.006). The weighted case has not been considered before