Crossing-Free Acyclic Hamiltonian Path Completion for Planar st-Digraphs

In this paper we study the problem of existence of a crossing-free acyclic hamiltonian path completion (for short, HP-completion) set for embedded upward planar digraphs. In the context of book embeddings, this question becomes: given an embedded upward planar digraph $G$, determine whether there exists an upward 2-page book embedding of $G$ preserving the given planar embedding. Given an embedded $st$-digraph $G$ which has a crossing-free HP-completion set, we show that there always exists a crossing-free HP-completion set with at most two edges per face of $G$. For an embedded $N$-free upward planar digraph $G$, we show that there always exists a crossing-free acyclic HP-completion set for $G$ which, moreover, can be computed in linear time. For a width-$k$ embedded planar $st$-digraph $G$, we show that we can be efficiently test whether $G$ admits a crossing-free acyclic HP-completion set.Comment: Accepted to ISAAC200

Upward Book Embeddings of st-Graphs

We study k-page upward book embeddings (kUBEs) of st-graphs, that is, book embeddings of single-source single-sink directed acyclic graphs on k pages with the additional requirement that the vertices of the graph appear in a topological ordering along the spine of the book. We show that testing whether a graph admits a kUBE is NP-complete for k >= 3. A hardness result for this problem was previously known only for k = 6 [Heath and Pemmaraju, 1999]. Motivated by this negative result, we focus our attention on k=2. On the algorithmic side, we present polynomial-time algorithms for testing the existence of 2UBEs of planar st-graphs with branchwidth b and of plane st-graphs whose faces have a special structure. These algorithms run in O(f(b)* n+n^3) time and O(n) time, respectively, where f is a singly-exponential function on b. Moreover, on the combinatorial side, we present two notable families of plane st-graphs that always admit an embedding-preserving 2UBE

Preprocessing for Outerplanar Vertex Deletion: An Elementary Kernel of Quartic Size

In the ?-Minor-Free Deletion problem one is given an undirected graph G, an integer k, and the task is to determine whether there exists a vertex set S of size at most k, so that G-S contains no graph from the finite family ? as a minor. It is known that whenever ? contains at least one planar graph, then ?-Minor-Free Deletion admits a polynomial kernel, that is, there is a polynomial-time algorithm that outputs an equivalent instance of size k^{?(1)} [Fomin, Lokshtanov, Misra, Saurabh; FOCS 2012]. However, this result relies on non-constructive arguments based on well-quasi-ordering and does not provide a concrete bound on the kernel size. We study the Outerplanar Deletion problem, in which we want to remove at most k vertices from a graph to make it outerplanar. This is a special case of ?-Minor-Free Deletion for the family ? = {K?, K_{2,3}}. The class of outerplanar graphs is arguably the simplest class of graphs for which no explicit kernelization size bounds are known. By exploiting the combinatorial properties of outerplanar graphs we present elementary reduction rules decreasing the size of a graph. This yields a constructive kernel with ?(k?) vertices and edges. As a corollary, we derive that any minor-minimal obstruction to having an outerplanar deletion set of size k has ?(k?) vertices and edges

Schematics of Graphs and Hypergraphs

Graphenzeichnen als ein Teilgebiet der Informatik befasst sich mit dem Ziel Graphen oder deren Verallgemeinerung Hypergraphen geometrisch zu realisieren. Beschränkt man sich dabei auf visuelles Hervorheben von wesentlichen Informationen in Zeichenmodellen, spricht man von Schemata. Hauptinstrumente sind Konstruktionsalgorithmen und Charakterisierungen von Graphenklassen, die für die Konstruktion geeignet sind. In dieser Arbeit werden Schemata für Graphen und Hypergraphen formalisiert und mit den genannten Instrumenten untersucht. In der Dissertation wird zunächst das „partial edge drawing“ (kurz: PED) Modell für Graphen (bezüglich gradliniger Zeichnung) untersucht. Dabei wird um Kreuzungen im Zentrum der Kante visuell zu eliminieren jede Kante durch ein kreuzungsfreies Teilstück (= Stummel) am Start- und am Zielknoten ersetzt. Als Standard hat sich eine PED-Variante etabliert, in der das Längenverhältnis zwischen Stummel und Kante genau 1⁄4 ist (kurz: 1⁄4-SHPED). Für 1⁄4-SHPEDs werden Konstruktionsalgorithmen, Klassifizierung, Implementierung und Evaluation präsentiert. Außerdem werden PED-Varianten mit festen Knotenpositionen und auf Basis orthogonaler Zeichnungen erforscht. Danach wird das BUS Modell für Hypergraphen untersucht, in welchem Hyperkanten durch fette horizontale oder vertikale – als BUS bezeichnete – Segmente repräsentiert werden. Dazu wird eine vollständige Charakterisierung von planaren Inzidenzgraphen von Hypergraphen angegeben, die eine planare Zeichnung im BUS Modell besitzen, und diverse planare BUS-Varianten mit festen Knotenpositionen werden diskutiert. Zum Schluss wird erstmals eine Punktmenge von subquadratischer Größe angegeben, die eine planare Einbettung (Knoten werden auf Punkte abgebildet) von 2-außenplanaren Graphen ermöglicht

Proceedings of the 8th Cologne-Twente Workshop on Graphs and Combinatorial Optimization

International audienceThe Cologne-Twente Workshop (CTW) on Graphs and Combinatorial Optimization started off as a series of workshops organized bi-annually by either Köln University or Twente University. As its importance grew over time, it re-centered its geographical focus by including northern Italy (CTW04 in Menaggio, on the lake Como and CTW08 in Gargnano, on the Garda lake). This year, CTW (in its eighth edition) will be staged in France for the first time: more precisely in the heart of Paris, at the Conservatoire National d’Arts et Métiers (CNAM), between 2nd and 4th June 2009, by a mixed organizing committee with members from LIX, Ecole Polytechnique and CEDRIC, CNAM

LIPIcs, Volume 248, ISAAC 2022, Complete Volume

LIPIcs, Volume 248, ISAAC 2022, Complete Volum