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 führen 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 müssen. Wir klassifizieren die Eingaben bezüglich ihrer Komplexität in Abhängigkeit 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 Komplexität bislang offen war. Insbesondere wird gezeigt, dass Clustered Level Planarity bereits für 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]. Außerdem wird eine Reduktion zu Manhattan Geodesic Planarity angegeben. Dies zeigt, dass ein bestehender [2009] Polynomialzeitalgorithmus für dieses Problem inkorrekt ist, es sei denn, dass P=NP ist. (2) Bucheinbettungen von dreifach zusammenhängenden planaren Graphen mit zwei Seiten: Wir zeigen, dass jeder dreifach zusammenhängende planare Graph mit Maximalgrad 5 Teilgraph eines Hamiltonischen planaren Graphen ist. Dies ist äquivalent dazu, dass ein solcher Graph eine Bucheinbettung auf zwei Seiten hat. Der Beweis ist konstruktiv und zeigt in der Tat sogar, dass es für die Realisierbarkeit nur notwendig ist, den Grad von Knoten separierender 3-Kreise zu beschränken - die übrigen Knoten können beliebig hohe Grade aufweisen. Dieses Ergebnis ist bestmöglich: Wenn die Gradschranke auf 6 abgeschwächt wird, gibt es Gegenbeispiele. Diese Ergebnisse verbessern Resultate von Heath [1995] und von Bauernöppel [1987] und, unabhängig davon, Bekos, Gronemann und Raftopoulou [2016], die gezeigt haben, dass planare Graphen mit Maximalgrad 3 beziehungsweise 4 auf zwei Seiten realisiert werden können. (3) Konvexitätssteigernde Deformationen: Wir zeigen, dass jede planare geradlinige Zeichnung eines intern dreifach zusammenhängenden planaren Graphen stetig zu einer solchen deformiert werden kann, in der jede Fläche ein konvexes Polygon ist. Dabei erhält die Deformation die Planarität und ist konvexitätssteigernd - 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

Packing and embedding large subgraphs

This thesis contains several embedding results for graphs in both random and non random settings. Most notably, we resolve a long standing conjecture that the threshold probability for Hamiltonicity in the random binomial subgraph of the hypercube equals $1/2$. %posed e.g.~by Bollob\'as, In Chapter 2 we obtain the following perturbation result regarding the hypercube \cQ^n: if H\subseteq\cQ^n satisfies $\delta(H)\geq\alpha n$ with $\alpha>0$ fixed and we consider a random binomial subgraph \cQ^n_p of \cQ^n with $p\in(0,1]$ fixed, then with high probability H\cup\cQ^n_p contains $k$ edge-disjoint Hamilton cycles, for any fixed $k\in\mathbb{N}$. This result is part of a larger volume of work where we also prove the corresponding hitting time result for Hamiltonicity. In Chapter 3 we move to a non random setting. %to a deterministic one. %Instead of embedding a single Hamilton cycle our result concerns packing more general families of graphs into a fixed host graph. Rather than pack a small number of Hamilton cycles into a fixed host graph, our aim is to achieve optimally sized packings of more general families of graphs. More specifically, we provide a degree condition on a regular $n$-vertex graph $G$ which ensures the existence of a near optimal packing of any family $\mathcal H$ of bounded degree $n$-vertex $k$-chromatic separable graphs into $G$. %In general, this degree condition is best possible. %In particular, this yields an approximate version of the tree packing conjecture %in the setting of regular host graphs $G$ of high degree. %Similarly, our result implies approximate versions of the Oberwolfach problem, %the Alspach problem and the existence of resolvable designs in the setting of %regular host graphs of high degree. In particular, this yields approximate versions of the the tree packing conjecture, the Oberwolfach problem, the Alspach problem and the existence of resolvable designs in the setting of regular host graphs of high degree

On some problems in extremal hypergraph theory

In this thesis, we study several problems in extremal (hyper)graph theory. We begin by investigating the problem of subgraph containment in the model of randomly perturbed graphs. In particular, we study the perturbed threshold for the appearance of the square of a Hamilton cycle and the problem of finding pairwise vertex-disjoint triangles. We provide a stability version of these results and we discuss their implications on the perturbed thresholds for 2-universality and for a triangle factor. We then turn to the notion of threshold in the context of transversals in hypergraph collections. Here the question is, given a fixed m-edge hypergraph F, how large the minimum degree of each hypergraph Hi needs to be, so that the hypergraph collection (H1, …, Hm) necessarily contains a transversal copy of F. We prove a widely applicable sufficient condition on F such that the following holds. The needed minimum degree is asymptotically the same as the minimum degree required for a copy of F to appear in each Hi. The condition is general enough to obtain transversal variants of various classical Dirac-type results for (powers of) Hamilton cycles. Finally, we initiate the study of a new variant of the Maker-Breaker positional game, which we call the (1:b) multistage Maker-Breaker game. Starting with a given hypergraph, we play several stages of a usual (1:b) Maker-Breaker game where, in each stage, we shrink the board by keeping only the elements that Maker claimed in the previous stage and updating the collection of winning sets accordingly. The game proceeds until no winning sets remain, and the goal of Maker is to prolong the duration of the game for as many stages as possible. We estimate the maximum duration of the (1:b) multistage Maker-Breaker game for several standard graph games played on the edge set of Kn with biases b subpolynomial in n

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum