Monotone Drawings of kk-Inner Planar Graphs

A $k$-inner planar graph is a planar graph that has a plane drawing with at most $k$ {internal vertices}, i.e., vertices that do not lie on the boundary of the outer face of its drawing. An outerplanar graph is a $0$-inner planar graph. In this paper, we show how to construct a monotone drawing of a $k$-inner planar graph on a $2(k+1)n \times 2(k+1)n$ grid. In the special case of an outerplanar graph, we can produce a planar monotone drawing on a $n \times n$ grid, improving previously known results.Comment: Appears in the Proceedings of the 26th International Symposium on Graph Drawing and Network Visualization (GD 2018). Revised introductio

Scratching the scale labyrinth

In this paper, we introduce a new approach to computer-aided microtonal improvisation by combining methods for (1) interactive scale navigation, (2) real-time manipulation of musical patterns and (3) dynamical timbre adaption in solidarity with the respective scales. On the basis of the theory of well-formed scales we offer a visualization of the underlying combinatorial ramifications in terms of a scale labyrinth. This involves the selection of generic well-formed scales on a binary tree (based on the Stern-Brocot tree) as well as the choice of specific tunings through the specification of the sizes of a period (pseudo-octave) and a generator (pseudo-fifth), whose limits are constrained by the actual position on the tree. We also introduce a method to enable transformations among the modes of a chosen scale (generalized and refined “diatonic” and “chromatic” transpositions). To actually explore the scales and modes through the shaping and transformation of rhythmically and melodically interesting tone patterns, we propose a playing technique called Fourier Scratching. It is based on the manipulation of the “spectra” (DFT) of playing gestures on a sphere. The coordinates of these gestures affect score and performance parameters such as scale degree, loudness, and timbre. Finally, we discuss a technique to dynamically match the timbre to the selected scale tuning

The combinatorics of plane curve singularities. How Newton polygons blossom into lotuses

This survey may be seen as an introduction to the use of toric and tropical geometry in the analysis of plane curve singularities, which are germs $(C,o)$ of complex analytic curves contained in a smooth complex analytic surface $S$. The embedded topological type of such a pair $(S, C)$ is usually defined to be that of the oriented link obtained by intersecting $C$ with a sufficiently small oriented Euclidean sphere centered at the point $o$, defined once a system of local coordinates $(x,y)$ was chosen on the germ $(S,o)$. If one works more generally over an arbitrary algebraically closed field of characteristic zero, one speaks instead of the combinatorial type of $(S, C)$. One may define it by looking either at the Newton-Puiseux series associated to $C$ relative to a generic local coordinate system $(x,y)$, or at the set of infinitely near points which have to be blown up in order to get the minimal embedded resolution of the germ $(C,o)$ or, thirdly, at the preimage of this germ by the resolution. Each point of view leads to a different encoding of the combinatorial type by a decorated tree: an Eggers-Wall tree, an Enriques diagram, or a weighted dual graph. The three trees contain the same information, which in the complex setting is equivalent to the knowledge of the embedded topological type. There are known algorithms for transforming one tree into another. In this paper we explain how a special type of two-dimensional simplicial complex called a lotus allows to think geometrically about the relations between the three types of trees. Namely, all of them embed in a natural lotus, their numerical decorations appearing as invariants of it. This lotus is constructed from the finite set of Newton polygons created during any process of resolution of $(C,o)$ by successive toric modifications.Comment: 104 pages, 58 figures. Compared to the previous version, section 2 is new. The historical information, contained before in subsection 6.2, is distributed now throughout the paper in the subsections called "Historical comments''. More details are also added at various places of the paper. To appear in the Handbook of Geometry and Topology of Singularities I, Springer, 202

Simple Compact Monotone Tree Drawings.

A monotone drawing of a graph G is a straight-line drawing of G such that every pair of vertices is connected by a path that is monotone with respect to some direction. Trees, as a special class of graphs, have been the focus of several papers and, recently, He and He [6] showed how to produce a monotone drawing of an arbitrary n-vertex tree that is contained in a 12n×12n grid. In this paper, we present a simple algorithm that constructs for each arbitrary tree a monotone drawing on a grid of size at most n×n