Many graphs arising in practice can be represented in a concise and intuitive way that conveys their structure. For example: A planar graph can be represented in the plane with points for vertices and non-crossing curves for edges. An interval graph can be represented on the real line with intervals for vertices and intersection of intervals representing edges. The concept of ``simultaneity'' applies for several types of graphs: the idea is to find representations for two graphs that share some common vertices and edges, and ensure that the common vertices and edges are represented the same way. Simultaneous representation problems arise in any situation where two related graphs should be represented consistently. A main instance is for temporal relationships, where an old graph and a new graph share some common parts. Pairs of related graphs arise in many other situations. For example, two social networks that share some members; two schedules that share some events, overlap graphs of DNA fragments of two similar organisms, circuit graphs of two adjacent layers on a computer chip etc. In this thesis, we study the simultaneous representation problem for several graph classes. For planar graphs the problem is defined as follows. Let G1 and G2 be two graphs sharing some vertices and edges. The simultaneous planar embedding problem asks whether there exist planar embeddings (or drawings) for G1 and G2 such that every vertex shared by the two graphs is mapped to the same point and every shared edge is mapped to the same curve in both embeddings. Over the last few years there has been a lot of work on simultaneous planar embeddings, which have been called `simultaneous embeddings with fixed edges'. A major open question is whether simultaneous planarity for two graphs can be tested in polynomial time. We give a linear-time algorithm for testing the simultaneous planarity of any two graphs that share a 2-connected subgraph. Our algorithm also extends to the case of k planar graphs, where each vertex [edge] is either common to all graphs or belongs to exactly one of them. Next we introduce a new notion of simultaneity for intersection graph classes (interval graphs, chordal graphs etc.) and for comparability graphs. For interval graphs, the problem is defined as follows. Let G1 and G2 be two interval graphs sharing some vertices I and the edges induced by I. G1 and G2 are said to be `simultaneous interval graphs' if there exist interval representations of G1 and G2 such that any vertex of I is assigned to the same interval in both the representations. The `simultaneous representation problem' for interval graphs asks whether G1 and G2 are simultaneous interval graphs. The problem is defined in a similar way for other intersection graph classes. For comparability graphs and any intersection graph class, we show that the simultaneous representation problem for the graph class is equivalent to a graph augmentation problem: given graphs G1 and G2, sharing vertices I and the corresponding induced edges, do there exist edges E' between G1-I and G2-I such that the graph G1 U G_2 U E' belongs to the graph class. This equivalence implies that the simultaneous representation problem is closely related to other well-studied classes in the literature, namely, sandwich graphs and probe graphs. We give efficient algorithms for solving the simultaneous representation problem for interval graphs, chordal graphs, comparability graphs and permutation graphs. Further, our algorithms for comparability and permutation graphs solve a more general version of the problem when there are multiple graphs, any two of which share the same common graph. This version of the problem also generalizes probe graphs.

Many graphs arising in practice can be represented in a concise and intuitive way that conveys their structure. For example: A planar graph can be represented in the plane with points for vertices and non-crossing curves for edges. An interval graph can be represented on the real line with intervals for vertices and intersection of intervals representing edges. The concept of ``simultaneity'' applies for several types of graphs: the idea is to find representations for two graphs that share some common vertices and edges, and ensure that the common vertices and edges are represented the same way. Simultaneous representation problems arise in any situation where two related graphs should be represented consistently. A main instance is for temporal relationships, where an old graph and a new graph share some common parts. Pairs of related graphs arise in many other situations. For example, two social networks that share some members; two schedules that share some events, overlap graphs of DNA fragments of two similar organisms, circuit graphs of two adjacent layers on a computer chip etc. In this thesis, we study the simultaneous representation problem for several graph classes. For planar graphs the problem is defined as follows. Let G1 and G2 be two graphs sharing some vertices and edges. The simultaneous planar embedding problem asks whether there exist planar embeddings (or drawings) for G1 and G2 such that every vertex shared by the two graphs is mapped to the same point and every shared edge is mapped to the same curve in both embeddings. Over the last few years there has been a lot of work on simultaneous planar embeddings, which have been called `simultaneous embeddings with fixed edges'. A major open question is whether simultaneous planarity for two graphs can be tested in polynomial time. We give a linear-time algorithm for testing the simultaneous planarity of any two graphs that share a 2-connected subgraph. Our algorithm also extends to the case of k planar graphs, where each vertex [edge] is either common to all graphs or belongs to exactly one of them. Next we introduce a new notion of simultaneity for intersection graph classes (interval graphs, chordal graphs etc.) and for comparability graphs. For interval graphs, the problem is defined as follows. Let G1 and G2 be two interval graphs sharing some vertices I and the edges induced by I. G1 and G2 are said to be `simultaneous interval graphs' if there exist interval representations of G1 and G2 such that any vertex of I is assigned to the same interval in both the representations. The `simultaneous representation problem' for interval graphs asks whether G1 and G2 are simultaneous interval graphs. The problem is defined in a similar way for other intersection graph classes. For comparability graphs and any intersection graph class, we show that the simultaneous representation problem for the graph class is equivalent to a graph augmentation problem: given graphs G1 and G2, sharing vertices I and the corresponding induced edges, do there exist edges E' between G1-I and G2-I such that the graph G1 U G_2 U E' belongs to the graph class. This equivalence implies that the simultaneous representation problem is closely related to other well-studied classes in the literature, namely, sandwich graphs and probe graphs. We give efficient algorithms for solving the simultaneous representation problem for interval graphs, chordal graphs, comparability graphs and permutation graphs. Further, our algorithms for comparability and permutation graphs solve a more general version of the problem when there are multiple graphs, any two of which share the same common graph. This version of the problem also generalizes probe graphs.

Relations between music education structures and teacher agency are under-researched and under-theorized, and scholars have indicated that the traditions and competitions of school bands in the U. S. may constrain educator agency. The need for research on teacher agency in competitive environments is compounded by policy trends toward administrators' use of festival scores in music educator evaluations. The purpose of this instrumental case study was to investigate band teacher agency in a highly competitive music education environment. I used the chordal triad of agency as the primary theoretical framework. Participants were four mid-career band educators in Texas, and I collected data through interviews, observations, journal entries, website review, and email correspondence. Throughout the data, participants' agency largely reproduced existing structures. Findings coalesced around (a) participants' core values of music, students' development, hard work, and competition, (b) an inductive, cohesive collection of band teaching norms despite participants' employment in schools of varying urbanicity and student demographics, (c) power sources that transmitted values and directed teachers' agency, and (d) a compelling story of one participant's generative agency that contrasted with the rest of the data. I provide directions for further research on music teacher agency and suggest implications for band educators, professional music education organizations, and music teacher educators.

Der Beitrag würdigt die Improvisation als eine die interpretatorischen und analytischen Verfahren ergänzende Perspektive des Werk- und Stilverstehens. Um die Erschrockenheit zu überbrücken, welche gemeinhin entsteht, wenn dem am Literaturspiel geschulten Instrumentalisten der sichere Boden des Notentextes entzogen wird, wird eine methodisch durchformte Improvisationsanleitung gegeben, die es ermöglicht, den neu gewonnenen Freiraum durch Begrenzung zu bewältigen, bevor es zum Überschreiten des zunächst gesetzten Rahmens kommt. Das Konzept gilt der tonalen Improvisation und orientiert sich an kompositorischen Modellen der Nach-Beethoven-Ära bis etwa 1850. Gleichwohl wird ein eher systematischer Ansatz verfolgt, der Übertragungen auch auf die Improvisation über Modelle der Wiener Klassik sowie eine Fortentwicklung zur Gruppenimprovisation, Akkordinstrument + Melodieinstrument(e)/Vokalstimme(n) gestattet. / This article addresses improvisation as a perspective complementary to interpretation and analysis in the explication of a work and its style. In order to bridge the shock that often results when an instrumentalist trained in playing literature leaves the safe haven of the printed score, a methodical guide to improvisation is offered that enables the player to conquer the newly won freedom by first introducing restrains before one eventually transgresses these initial boundaries. The method addresses tonal improvisation and is oriented around compositional models of the time following Beethoven, ca. 1850. At the same time a rather systematic approach is employed that allows crossover into improvisation over models drawn from Viennese classicism as well as extension activities involving group-improvisation and the combination of chordal instruments, melody instruments, and voice.

This PhD thesis is comprised of 3 results in computational geometry and graph theory. In the first paper, I demonstrate that the piercing number of a set S of pairwise intersecting convex shapes in the plane is bounded by O(\alpha(S)), where \alpha(S) is the fatness of the set S, improving upon the previous upper-bound of O(\alpha(S)^2). In the second article, I show that anagram-free vertex colouring of a 2\times n square grid requires a number of colours that increases with n. This answers an open question in Wilson's thesis and shows that even graphs of pathwidth 2 do not have anagram-free colouring with a bounded number of colours. The third article is a study on the geodesic anagram-free chromatic number of chordal and interval graphs. \emph{Geodesic anagram-free chromatic number} is defined as the minimum number of colours required to colour a graph such that all shortest paths between any pair of vertices are coloured anagram-free. In particular, I prove that the geodesic anagram-free chromatic number of a chordal graph G is 32p'w, where p' is the pathwidth of the subtree intersection representation graph (tree) of G, and w is the clique number of G. Additionally, I prove that the geodesic anagram-free chromatic number of an interval graph is bounded by 32p, where p is the pathwidth of the interval graph. This PhD thesis is comprised of 3 results in computational geometry and graph theory.

No description available.

Diese Arbeit leistet Beiträge im Bereich der deskriptiven Komplexitätstheorie. Zunächst beschäftigen wir uns mit der ungelösten Frage, ob es eine Logik gibt, welche die Klasse der Polynomialzeit-Eigenschaften (PTIME) charakterisiert. Wir betrachten Graphklassen, die unter induzierten Teilgraphen abgeschlossen sind. Auf solchen Graphklassen lässt sich die 1976 von Gallai eingeführte modulare Zerlegung anwenden. Graphen, die durch modulare Zerlegung nicht zerlegbar sind, heißen prim. Wir stellen ein neues Werkzeug vor: das Modulare Zerlegungstheorem. Es reduziert (definierbare) Kanonisierung einer Graphklasse C auf (definierbare) Kanonisierung der Klasse aller primen Graphen aus C, die mit binären Relationen auf einer linear geordneten Menge gefärbt sind. Mit Hilfe des Modularen Zerlegungstheorems zeigen wir, dass Fixpunktlogik mit Zählen (FP+C) PTIME auf der Klasse aller Permutationsgraphen und auf der Klasse aller chordalen Komparabilitätsgraphen charakterisiert. Wir beweisen zudem, dass modulare Zerlegungsbäume in Symmetrisch-Transitive-Hüllen-Logik mit Zählen (STC+C) definierbar und damit in logarithmischem Platz berechenbar sind. Weiterhin definieren wir eine neue Logik für die Komplexitätsklasse Logarithmischer Platz (LOGSPACE). Wir erweitern die Logik erster Stufe mit Zählen um einen Operator, der eine in logarithmischem Platz berechenbare Form der Rekursion erlaubt. Die resultierende Logik LREC ist ausdrucksstärker als die Deterministisch-Transitive-Hüllen-Logik mit Zählen (DTC+C) und echt in FP+C enthalten. Wir zeigen, dass LREC LOGSPACE auf gerichteten Bäumen charakterisiert. Zudem betrachten wir eine Erweiterung LREC= von LREC, die sich gegenüber LREC durch bessere Abschlusseigenschaften auszeichnet und im Gegensatz zu LREC ausdrucksstärker als die Symmetrisch-Transitive-Hüllen-Logik (STC) ist. Wir beweisen, dass LREC= LOGSPACE sowohl auf der Klasse der Intervallgraphen als auch auf der Klasse der chordalen klauenfreien Graphen charakterisiert. / This theses is making contributions to the field of descriptive complexity theory. First, we look at the main open problem in this area: the question of whether there exists a logic that captures polynomial time (PTIME). We consider classes of graphs that are closed under taking induced subgraphs. For such graph classes, an effective graph decomposition, called modular decomposition, was introduced by Gallai in 1976. The graphs that are non-decomposable with respect to modular decomposition are called prime. We present a tool, the Modular Decomposition Theorem, that reduces (definable) canonization of a graph class C to (definable) canonization of the class of prime graphs of C that are colored with binary relations on a linearly ordered set. By an application of the Modular Decomposition Theorem, we show that fixed-point logic with counting (FP+C) captures PTIME on the class of permutation graphs and the class of chordal comparability graphs. We also prove that the modular decomposition tree is definable in symmetric transitive closure logic with counting (STC+C), and therefore, computable in logarithmic space. Further, we introduce a new logic for the complexity class logarithmic space (LOGSPACE). We extend first-order logic with counting by a new operator that allows it to formalize a limited form of recursion which can be evaluated in logarithmic space. We prove that the resulting logic LREC is strictly more expressive than deterministic transitive closure logic with counting (DTC+C) and that it is strictly contained in FP+C. We show that LREC captures LOGSPACE on the class of directed trees. We also study an extension LREC= of LREC that has nicer closure properties and that, unlike LREC, is more expressive than symmetric transitive closure logic (STC). We prove that LREC= captures LOGSPACE on the class of interval graphs and on the class of chordal claw-free graphs.

Nous avons utilisé la décomposition par séparateurs minimaux complets. Pour décomposer un graphe G, il est nécessaire de trouver les séparateurs minimaux dans le graphe triangulé H correspondant. Dans ce contexte, nos premiers efforts se sont tournés vers la détection de séparateurs minimaux dans un graphe triangulé. Nous avons défini une structure, que nous avons nommée 'atom tree'. Cette dernière est inspirée du 'clique tree' et permet d'obtenir et de représenter les atomes qui sont les produits de la décomposition. Lors de la manipulation de données à l'aide de treillis de Galois, nous avons remarqué que la décomposition par séparateurs minimaux permettait une approche de type `Diviser pour régner' pour les treillis de Galois. La détection des gènes fusionnés, qui est une étape importante pour la compréhension de l'évolution des espèces, nous a permis d'appliquer nos algorithmes de détection de séparateurs minimaux complets, qui nous a permis de détecter et regrouper de manière efficace les gènes fusionnés. Une autre application biologique fut la détection de familles de gènes d'intérêts à partir de données de niveaux d'expression de gènes. La structure de `l'atom tree' nous a permis d'avoir un bon outils de visualisation et de gérer des volumes de données importantes. / We worked on clique minimal separator decomposition. In order to compute this decomposition on a graph G we need to compute the minimal separators of its triangulation H. In this context, the first efforts were on finding a clique minimal separators in a chordal graph. We defined a structure called atom tree inspired from the clique tree to compute and represent the final products of the decomposition, called atoms. The purpose of this thesis was to apply this technique on biological data. While we were manipulating this data using Galois lattices, we noticed that the clique minimal separator decomposition allows a divide and conquer approach on Galois lattices. One biological application of this thesis was the detection of fused genes which are important evolutionary events. Using algorithms we produced in the course of along our work we implemented a program called MosaicFinder that allows an efficient detection of this fusion event and their pooling. Another biological application was the extraction of genes of interest using expression level data. The atom tree structure allowed us to have a good visualization of the data and to be able to compute large datasets.