Clique graphs and Helly graphs

AbstractAmong the graphs for which the system of cliques has the Helly property those are characterized which are clique-convergent to the one-vertex graph. These graphs, also known as the so-called absolute retracts of reflexive graphs, are the line graphs of conformal Helly hypergraphs possessing a certain elimination scheme. From particular classes of such hypergraphs one can readily construct various classes G of graphs such that each member of G has its clique graph in G and is itself the clique graph of some other member of G. Examples include the classes of strongly chordal graphs and Ptolemaic graphs, respectively

Modellbildung versus Modellisieren und Scheinmodellierung

Was ist mathematische Modellierung? Diese Frage wird Semester um Semester an vielen Universitäten zu Beginn einer Vorlesung, die ,Modellierung‘ im Titel trägt, rhetorisch gestellt. Dabei sind meist Dynamische Systeme zur Analyse realer zeitabhängiger Prozesse in den Naturwissenschaften und der Technik (aber nicht ausschließlich) Gegenstand der Untersuchung [...]

Absolute reflexive retracts and absolute bipartite retracts

AbstractIt is a well-known phenomenon in the study of graph retractions that most results about absolute retracts in the class of bipartite (irreflexive) graphs have analogues about absolute retracts in the class of reflexive graphs, and vice versa. In this paper we make some observations that make the connection explicit. We develop four natural transformations between reflexive graphs and bipartite graphs which preserve the property of being an absolute retract, and allow us to derive results about absolute reflexive retracts from similar results about absolute bipartite retracts and conversely. Then we introduce generic notions that specialize to the appropriate concepts in both cases. This paves the way to a unified view of both theories, leading to absolute retracts of general (i.e., partially reflexive) graphs

Combinatorics and geometry of finite and infinite squaregraphs

Squaregraphs were originally defined as finite plane graphs in which all inner faces are quadrilaterals (i.e., 4-cycles) and all inner vertices (i.e., the vertices not incident with the outer face) have degrees larger than three. The planar dual of a finite squaregraph is determined by a triangle-free chord diagram of the unit disk, which could alternatively be viewed as a triangle-free line arrangement in the hyperbolic plane. This representation carries over to infinite plane graphs with finite vertex degrees in which the balls are finite squaregraphs. Algebraically, finite squaregraphs are median graphs for which the duals are finite circular split systems. Hence squaregraphs are at the crosspoint of two dualities, an algebraic and a geometric one, and thus lend themselves to several combinatorial interpretations and structural characterizations. With these and the 5-colorability theorem for circle graphs at hand, we prove that every squaregraph can be isometrically embedded into the Cartesian product of five trees. This embedding result can also be extended to the infinite case without reference to an embedding in the plane and without any cardinality restriction when formulated for median graphs free of cubes and further finite obstructions. Further, we exhibit a class of squaregraphs that can be embedded into the product of three trees and we characterize those squaregraphs that are embeddable into the product of just two trees. Finally, finite squaregraphs enjoy a number of algorithmic features that do not extend to arbitrary median graphs. For instance, we show that median-generating sets of finite squaregraphs can be computed in polynomial time, whereas, not unexpectedly, the corresponding problem for median graphs turns out to be NP-hard.Comment: 46 pages, 14 figure

Approximation algorithms for multi-dimensional assignment problems with decomposable costs

AbstractThe k-dimensional assignment problem with decomposable costs is formulated as follows. Given is a complete k-partite graph G = (X0 ∪ ⋯ ∪ Xk − 1, E), with |Xi| = p for each i, and a nonnegative length function defined on the edges of G. A clique of G is a subset of vertices meeting each Xi in exactly one vertex. The cost of a clique is a function of the lengths of the edges induced by the clique. Four specific cost functions are considered in this paper; namely, the cost of a clique is either the sum of the lengths of the edges induced by the clique (sum costs), or the minimum length of a spanning star (star costs) or of a traveling salesman tour (tour costs) or of a spanning tree (tree costs) of the induced subgraph. The problem is to find a minimum-cost partition of the vertex set of G into cliques. We propose several simple heuristics for this problem, and we derive worst-case bounds on the ratio between the cost of the solutions produced by these heuristics and the cost of an optimal solution. The worst-case bounds are stated in terms of two parameters, viz. k and τ, where the parameter τ indicates how close the edge length function comes to satisfying the triangle inequality

Schöne neue Mathewelt der österreichischen Zentralmatura 2015

Die Aufgaben zur Mathematik in der neuen „standardisierten kompetenzorientierten schriftlichen Reifeprüfung 2015“ in Österreich werden kritisch betrachtet. Besonderes Augenmerk wird darauf gelegt, welchen Klassenstufen sie zuzuordnen sind und ab welcher Klassenstufe man diese Klausur folglich bestehen kann. Es zeigt sich, dass schon die ersten 9-10 Schuljahre dafür ausreichen. Dies lässt nur den Schluss zu, dass auf diese Weise letztlich beabsichtigt ist, beim Fach Mathematik die Matura auf das Niveau des in Deutschland so genannten „mittleren Schulabschlusses“ nach 10 Schuljahren abzusenken, den es in Österreich in dieser Form allerdings nicht gibt

Noch einmal: Schöne neue Mathewelt

Unser Aufsatz „Schöne neue Mathewelt“ in den Mitteilungen der GDM Band 100 erfuhr in zweiBeiträgen eine kritische Antwort in Band 101 (Dorner  Götz 2016; Bruder, Linnemann, Sattlberger, Siller Steinfeld 2016). In diesem Beitrag wird auf zwei zentrale Kritikpunkte aus diesen Reaktionen eingegangen

Die Selbstunterwerfung unter ökonomisches Denken.

Die Autoren zeigen, wie die Kompetenzorientierung an Schulen und deren Forcierung über die Bildungsforschung zu einer Ökonomisierung des Bildungsbereiches beiträgt. Am Mathematikunterricht können sie nachweisen, dass durch die Fokussierung auf Kompetenzen vor allem eine ökonomische Haltung zu den Dingen bei den Schülerinnen und Schülern eingeübt wird, nicht aber am Verstehen der mathematischen Inhalte mit ihnen gearbeitet wird. Kompetenz ist demzufolge nicht mehr an der Sache, sondern anderen bloßer Verwertung ausgerichtet. (DIPF/sj