Classification is an important task at which both biological and artificial neural networks excel 1,2 . In machine learning, nonlinear projection into a high-dimensional feature space can make data linearly separable 3,4 , simplifying the classification of complex features. Such nonlinear projections are computationally expensive in conventional computers. A promising approach is to exploit physical materials systems that perform this nonlinear projection intrinsically, because of their high computational density 5 , inherent parallelism and energy efficiency 6,7 . However, existing approaches either rely on the systems' time dynamics, which requires sequential data processing and therefore hinders parallel computation 5,6,8 , or employ large materials systems that are difficult to scale up 7 . Here we use a parallel, nanoscale approach inspired by filters in the brain 1 and artificial neural networks 2 to perform nonlinear classification and feature extraction. We exploit the nonlinearity of hopping conduction [9][10][11] through an electrically tunable network of boron dopant atoms in silicon, reconfiguring the network through artificial evolution to realize different computational functions. We first solve the canonical two-input binary classification problem, realizing all Boolean logic gates 12 up to room temperature, demonstrating nonlinear classification with the nanomaterial system. We then evolve our dopant network to realize feature filters 2 that can perform four-input binary classification on the Modified National Institute of Standards and Technology handwritten digit database. Implementation of our material-based filters substantially improves the classification accuracy over that of a linear classifier directly applied to the original data 13 . Our results establish a paradigm of silicon-based electronics for smallfootprint and energy-efficient computation 14 .Doping is a crucial process in semiconductor electronics, where impurity atoms are introduced to modulate the charge carrier concentration. Conventional semiconductor devices operate in the band regime of charge transport, where the delocalization of the charge carriers gives rise to high mobility and a linear response to an applied electric field. At sufficiently low doping concentration and temperature 9,15 , however, delocalization is lost, and carriers move sequentially from dopant atom to dopant atom. This is referred to as the hopping regime 10,11,16 , which exhibits higher resistivity and nonlinearity. Nonlinearity is often undesired, but it is a valuable asset for unconventional computing, that is, for systems that do not follow the Turing model of computation [6][7][8][17][18][19] . Rather than excluding nonlinearity, we can exploit it 12 and manipulate our physical system with artificial evolution to solve computational problems 17 . This evolution in materio has been used, for example, for frequency distinguishing by liquid crystals 18 and robot control with carbon nanotubes 19 . We recently showed that a disordered network of gold...

Natural computers exploit the emergent properties and massive parallelism of interconnected networks of locally active components. Evolution has resulted in systems that compute quickly and that use energy efficiently, utilizing whatever physical properties are exploitable. Man-made computers, on the other hand, are based on circuits of functional units that follow given design rules. Hence, potentially exploitable physical processes, such as capacitive crosstalk, to solve a problem are left out. Until now, designless nanoscale networks of inanimate matter that exhibit robust computational functionality had not been realized. Here we artificially evolve the electrical properties of a disordered nanomaterials system (by optimizing the values of control voltages using a genetic algorithm) to perform computational tasks reconfigurably. We exploit the rich behaviour that emerges from interconnected metal nanoparticles, which act as strongly nonlinear single-electron transistors, and find that this nanoscale architecture can be configured in situ into any Boolean logic gate. This universal, reconfigurable gate would require about ten transistors in a conventional circuit. Our system meets the criteria for the physical realization of (cellular) neural networks: universality (arbitrary Boolean functions), compactness, robustness and evolvability, which implies scalability to perform more advanced tasks. Our evolutionary approach works around device-to-device variations and the accompanying uncertainties in performance. Moreover, it bears a great potential for more energy-efficient computation, and for solving problems that are very hard to tackle in conventional architectures.

We present (9 4 −)-tough graphs without a Hamilton path for arbitrary ¿ 0, thereby refuting a well-known conjecture due to Chvà atal. We also present (7 4 −)-tough chordal graphs without a Hamilton path for any ¿ 0.

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The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-pro t purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. AbstractWe prove three complexity results on vertex coloring problems restricted to P k -free graphs, i.e., graphs that do not contain a path on k vertices as an induced subgraph. First of all, we show that the pre-coloring extension version of 5-coloring remains NP-complete when restricted to P 6 -free graphs. Recent results of Hoàng et al. imply that this problem is polynomially solvable on P 5 -free graphs. Secondly, we show that the pre-coloring extension version of 3-coloring is polynomially solvable for P 6 -free graphs. This implies a simpler algorithm for checking the 3-colorability of P 6 -free graphs than the algorithm given by Randerath and Schiermeyer. Finally, we prove that 6-coloring is NP-complete for P 7 -free graphs. This problem was known to be polynomially solvable for P 5 -free graphs and NP-complete for P 8 -free graphs, so there remains one open case.

Given a graph G = (V, E) and a (not necessarily proper) edgecoloring of G, we consider the complexity of finding a spanning tree of G with as many different colors as possible, and of finding one with as few different colors as possible. We show that the first problem is equivalent to finding a common independent set of maximum cardinality in two matroids, implying that there is a polynomial algorithm. We use the minimum dominating set problem to show that the second problem is N P -hard.

The concept of a line graph is generalized to that of a path graph. The path graph f,(G) of a graph G is obtained by representing the paths Pk in G by vertices and joining two vertices whenever the corresponding paths f k in G form a path f k + , or a cycle C,. f,-graphs are characterized and investigated on isomorphism and traversability. Trees and unicyclic graphs with hamiltonian /?,-graphs are characterized.