Cutting up graphs revisited - a short proof of Stallings' structure theorem

This is a new and short proof of the main theorem of classical structure tree theory. Namely, we show the existence of certain automorphism-invariant tree-decompositions of graphs based on the principle of removing finitely many edges. This was first done in "Cutting up graphs" by M.J. Dunwoody. The main ideas are based on the paper "Vertex cuts" by M.J. Dunwoody and the author. We extend the theorem to a detailed combinatorial proof of J.R. Stallings' theorem on the structure of finitely generated groups with more than one end.Comment: 12 page

Bin Packing and Related Problems: General Arc-flow Formulation with Graph Compression

We present an exact method, based on an arc-flow formulation with side constraints, for solving bin packing and cutting stock problems --- including multi-constraint variants --- by simply representing all the patterns in a very compact graph. Our method includes a graph compression algorithm that usually reduces the size of the underlying graph substantially without weakening the model. As opposed to our method, which provides strong models, conventional models are usually highly symmetric and provide very weak lower bounds. Our formulation is equivalent to Gilmore and Gomory's, thus providing a very strong linear relaxation. However, instead of using column-generation in an iterative process, the method constructs a graph, where paths from the source to the target node represent every valid packing pattern. The same method, without any problem-specific parameterization, was used to solve a large variety of instances from several different cutting and packing problems. In this paper, we deal with vector packing, graph coloring, bin packing, cutting stock, cardinality constrained bin packing, cutting stock with cutting knife limitation, cutting stock with binary patterns, bin packing with conflicts, and cutting stock with binary patterns and forbidden pairs. We report computational results obtained with many benchmark test data sets, all of them showing a large advantage of this formulation with respect to the traditional ones

Characterization and Defect Analysis of Machined Regions in Al-SiC Metal Matrix Composites Using an Abrasive Water Jet Machining Process

Metal matrix composite (MMC) materials are increasingly used in industrial sectors such as energy, structural, aerospace, and automotive. This is due to the improvement of properties by the addition of reinforcements. Thus, it is possible to obtain properties of higher strength, better rigidity, controlled thermal expansion, and elevated wear resistance. However, due to the extreme hardness achieved during their manufacture, these composites pose a challenge to the conventional machining industry due to the rapid deterioration experienced by cutting tools. This article therefore proposes the use of an unconventional machining method that is becoming increasingly widely used: abrasive water jet cutting. This process is characterized by high production rates, absence of wear, and environmental friendliness, among other advantages. Experimental tests were carried out in order to analyze results that minimize the formation of defects in the machining of metal matrix composite consisting of aluminium matrix with silicon carbide (Al-SiC MMC). To this end, results were analyzed using Scanning Optical and Electron Microscope (SOM/SEM) techniques, the taper angle was calculated, and areas with different surface quality were detected by measuring the roughness

4-colored graphs and knot/link complements

A representation for compact 3-manifolds with non-empty non-spherical boundary via 4-colored graphs (i.e., 4-regular graphs endowed with a proper edge-coloration with four colors) has been recently introduced by two of the authors, and an initial classification of such manifolds has been obtained up to 8 vertices of the representing graphs. Computer experiments show that the number of graphs/manifolds grows very quickly as the number of vertices increases. As a consequence, we have focused on the case of orientable 3-manifolds with toric boundary, which contains the important case of complements of knots and links in the 3-sphere. In this paper we obtain the complete catalogation/classification of these 3-manifolds up to 12 vertices of the associated graphs, showing the diagrams of the involved knots and links. For the particular case of complements of knots, the research has been extended up to 16 vertices.Comment: 19 pages, 6 figures, 3 tables; changes in Lemma 6, Corollaries 7 and

Modeling and optimization of surface roughness and vibration amplitude in heat assisted end milling of SKD 11 tool steel using ball nose tool

Tool steel - SKD 11 is frequently used in industries for making dies and molds. This grade is chosen for its toughness, strength, and hardness maintained up to high temperature. However, the same properties make the steel extremely difficult and expensive to machine using conventional approaches. Heat assisted machining has been found wide spread application in recent years to improve machinability of difficult-to-cut materials. This research paper presents the outcome of an investigation on heat assisted end milling of SKD 11 conducted on a vertical machining center using ball nose coated carbide inserts. The Design of Experiments (DoE) was done using the Response Surface Methodology, in order to develop empirical mathematical models of surface roughness and vibration in terms of cutting speed, feed, axial depth of cut, and heating temperature. The models were checked for significance using Analysis of Variance (ANOVA). 3-D response surface graphs of the interactions of primary cutting parameters with the responses were plotted. Optimization was then performed by using the desirability function approach. From the graphs and optimized results it was concluded that the primary input parameters could be controlled in order to reduce vibration amplitude and produce semi-finished machined surfaces applying induction heat assisted technique