Efficient domination and polarity

The thesis considers the following graph problems: Efficient (Edge) Domination seeks for an independent vertex (edge) subset D such that all other vertices (edges) have exactly one neighbor in D. Polarity asks for a vertex subset that induces a complete multipartite graph and that contains a vertex of every induced P_3. Monopolarity is the special case of Polarity where the wanted vertex subset has to be independent. These problems are NP-complete in general, but efficiently solvable on various graph classes. The thesis sharpens known NP-completeness results and presents new solvable cases

Upper density problems in infinite Ramsey theory

We consider the following question in infinite Ramsey theory, introduced by Erdős and Galvin [EG93] in a particular case and by DeBiasio and McKenney [DM19] in a more general setting. Let H be a countably infinite graph. If the edges of the complete graph on the natural numbers are colored red or blue, what is the maximum value of λ such that we are guaranteed to find a monochromatic copy of H whose vertex set has upper density at least λ? We call this value the Ramsey density of H. The problem of determining the Ramsey density of the infinite path was first studied by Erdős and Galvin, and was recently solved by Corsten, DeBiasio, Lang and the author [CDLL19]. In this thesis we study the problem of determining the Ramsey density of arbitrary graphs H. On an intuitive level, we show that three properties of a graph H have an effect on the Ramsey density: the chromatic number, the number of components, and the expansion of its independent sets. We deduce the exact value of the Ramsey density for a wide variety of graphs, including all locally finite forests, bipartite factors, clique factors and odd cycle factors. We also determine the value of the Ramsey density of all locally finite graphs, up to a factor of 2. We also study a list coloring variant of the same problem. We show that there exists a way of assigning a list of size two to every edge in the complete graph on N such that, in every list coloring, there are monochromatic paths with density arbitrarily close to 1.Wir betrachten die folgende Fragestellung aus der Ramsey-Theorie, welche von Erdős und Galvin [EG93] in einem Spezialfall sowie von DeBiasio und McKenney [DM19] in einem allgemeineren Kontext formuliert wurde: Es sei H ein abzählbar unendlicher Graph. Welches ist der größtmögliche Wert λ, sodass wir, wenn die Kanten des vollständigen Graphen mit Knotenmenge N jeweils entweder rot oder blau gefärbt sind, stets eine einfarbige Kopie von H, dessen Knotenmenge eine obere asymptotische Dichte von mindestens λ besitzt, finden können? Wir nennen diesen Wert die Ramsey-Dichte von H. Das Problem, die Ramsey-Dichte des unendlichen Pfades zu bestimmen wurde erstmals von Erdős und Galvin untersucht und wurde vor kurzem von Corsten, DeBiasio, Lang und dem Autor [CDLL19] gelöst. Gegenstand der vorliegenden Dissertation ist die Bestimmung der Ramsey-Dichten von Graphen. Auf einer intuitiven Ebene zeigen wir, dass drei Parameter eines Graphen die Ramsey-Dichte beeinflussen: die chromatische Zahl, die Anzahl der Zusammenhangskomponenten sowie die Expansion seiner unabhängigen Mengen. Wir ermitteln die exakten Werte der Ramsey-Dichte für eine Vielzahl von Graphen, darunter alle lokal endlichen Wälder, bipartite Faktoren, Kr-Faktoren sowie Ck-Faktoren für ungerade k. Ferner bestimmen wir den Wert der Ramsey-Dichte aller lokal endlichen Graphen bis auf einen Faktor 2. Darüber hinaus untersuchen wir eine Variante des oben beschriebenen Problems für Listenfärbungen. Wir zeigen, dass es möglich ist, jeder Kante des vollständigen Graphen mit Knotenmenge N eine Liste der Größe Zwei zuzuweisen, sodass in jeder zugehörigen Listenfärbung monochromatische Pfade mit beliebig nah an 1 liegender Dichte existieren

Caching Connections in Matchings

Motivated by the desire to utilize a limited number of configurable optical switches by recent advances in Software Defined Networks (SDNs), we define an online problem which we call the Caching in Matchings problem. This problem has a natural combinatorial structure and therefore may find additional applications in theory and practice. In the Caching in Matchings problem our cache consists of $k$ matchings of connections between servers that form a bipartite graph. To cache a connection we insert it into one of the $k$ matchings possibly evicting at most two other connections from this matching. This problem resembles the problem known as Connection Caching, where we also cache connections but our only restriction is that they form a graph with bounded degree $k$. Our results show a somewhat surprising qualitative separation between the problems: The competitive ratio of any online algorithm for caching in matchings must depend on the size of the graph. Specifically, we give a deterministic $O(nk)$ competitive and randomized $O(n \log k)$ competitive algorithms for caching in matchings, where $n$ is the number of servers and $k$ is the number of matchings. We also show that the competitive ratio of any deterministic algorithm is $\Omega(\max(\frac{n}{k},k))$ and of any randomized algorithm is $\Omega(\log \frac{n}{k^2 \log k} \cdot \log k)$. In particular, the lower bound for randomized algorithms is $\Omega(\log n)$ regardless of $k$, and can be as high as $\Omega(\log^2 n)$ if $k=n^{1/3}$, for example. We also show that if we allow the algorithm to use at least $2k-1$ matchings compared to $k$ used by the optimum then we match the competitive ratios of connection catching which are independent of $n$. Interestingly, we also show that even a single extra matching for the algorithm allows to get substantially better bounds

Boundary properties of graphs

A set of graphs may acquire various desirable properties, if we apply suitable restrictions on the set. We investigate the following two questions: How far, exactly, must one restrict the structure of a graph to obtain a certain interesting property? What kind of tools are helpful to classify sets of graphs into those which satisfy a property and those that do not? Equipped with a containment relation, a graph class is a special example of a partially ordered set. We introduce the notion of a boundary ideal as a generalisation of a notion introduced by Alekseev in 2003, to provide a tool to indicate whether a partially ordered set satisfies a desirable property or not. This tool can give a complete characterisation of lower ideals defined by a finite forbidden set, into those that satisfy the given property and to those that do not. In the case of graphs, a lower ideal with respect to the induced subgraph relation is known as a hereditary graph class. We study three interrelated types of properties for hereditary graph classes: the existence of an efficient solution to an algorithmic graph problem, the boundedness of the graph parameter known as clique-width, and well-quasi-orderability by the induced subgraph relation. It was shown by Courcelle, Makowsky and Rotics in 2000 that, for a graph class, boundedness of clique-width immediately implies an efficient solution to a wide range of algorithmic problems. This serves as one of the motivations to study clique-width. As for well-quasiorderability, we conjecture that every hereditary graph class that is well-quasi-ordered by the induced subgraph relation also has bounded clique-width. We discover the first boundary classes for several algorithmic graph problems, including the Hamiltonian cycle problem. We also give polynomial-time algorithms for the dominating induced matching problem, for some restricted graph classes. After discussing the special importance of bipartite graphs in the study of clique-width, we describe a general framework for constructing bipartite graphs of large clique-width. As a consequence, we find a new minimal class of unbounded clique-width. We prove numerous positive and negative results regarding the well-quasi-orderability of classes of bipartite graphs. This completes a characterisation of the well-quasi-orderability of all classes of bipartite graphs defined by one forbidden induced bipartite subgraph. We also make considerable progress in characterising general graph classes defined by two forbidden induced subgraphs, reducing the task to a small finite number of open cases. Finally, we show that, in general, for hereditary graph classes defined by a forbidden set of bounded finite size, a similar reduction is not usually possible, but the number of boundary classes to determine well-quasi-orderability is nevertheless finite. Our results, together with the notion of boundary ideals, are also relevant for the study of other partially ordered sets in mathematics, such as permutations ordered by the pattern containment relation

The world of hereditary graph classes viewed through Truemper configurations

In 1982 Truemper gave a theorem that characterizes graphs whose edges can be labeled so that all chordless cycles have prescribed parities. The characterization states that this can be done for a graph G if and only if it can be done for all induced subgraphs of G that are of few speci c types, that we will call Truemper con gurations. Truemper was originally motivated by the problem of obtaining a co-NP characterization of bipartite graphs that are signable to be balanced (i.e. bipartite graphs whose node-node incidence matrices are balanceable matrices). The con gurations that Truemper identi ed in his theorem ended up playing a key role in understanding the structure of several seemingly diverse classes of objects, such as regular matroids, balanceable matrices and perfect graphs. In this survey we view all these classes, and more, through the excluded Truemper con gurations, focusing on the algorithmic consequences, trying to understand what structurally enables e cient recognition and optimization algorithms

Solving Problems on Graphs of High Rank-Width

A modulator of a graph G to a specified graph class H is a set of vertices whose deletion puts G into H. The cardinality of a modulator to various tractable graph classes has long been used as a structural parameter which can be exploited to obtain FPT algorithms for a range of hard problems. Here we investigate what happens when a graph contains a modulator which is large but "well-structured" (in the sense of having bounded rank-width). Can such modulators still be exploited to obtain efficient algorithms? And is it even possible to find such modulators efficiently? We first show that the parameters derived from such well-structured modulators are strictly more general than the cardinality of modulators and rank-width itself. Then, we develop an FPT algorithm for finding such well-structured modulators to any graph class which can be characterized by a finite set of forbidden induced subgraphs. We proceed by showing how well-structured modulators can be used to obtain efficient parameterized algorithms for Minimum Vertex Cover and Maximum Clique. Finally, we use well-structured modulators to develop an algorithmic meta-theorem for deciding problems expressible in Monadic Second Order (MSO) logic, and prove that this result is tight in the sense that it cannot be generalized to LinEMSO problems.Comment: Accepted at WADS 201

Coloring, List Coloring, and Painting Squares of Graphs (and other related problems)

We survey work on coloring, list coloring, and painting squares of graphs; in particular, we consider strong edge-coloring. We focus primarily on planar graphs and other sparse classes of graphs.Comment: 32 pages, 13 figures and tables, plus 195-entry bibliography, comments are welcome, published as a Dynamic Survey in Electronic Journal of Combinatoric