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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.

A well‐known formula of Tutte and Berge expresses the size of a maximum matching in a graph Gin terms of what is usually called the deficiency of G. A subset Xof V(G) for which this deficiency is attained is called a Tutte set of G. While much is known about maximum matchings, less is known about the structure of Tutte sets. In this article, we study the structural aspects of maximal Tutte sets in a graph G. Towards this end, we introduce a related graph D(G). We first show that the maximal Tutte sets in Gare precisely the maximal independent sets in its D‐graph D(G), and then continue with the study of D‐graphs in their own right, and of iterated D‐graphs. We show that Gis isomorphic to a spanning subgraph of D(G), and characterize the graphs for which G≅D(G) and for which D(G)≅D2(G). Surprisingly, it turns out that for every graph Gwith a perfect matching, D3(G)≅D2(G). Finally, we characterize bipartite D‐graphs and comment on the problem of characterizing D‐graphs in general. © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 343–358, 2007

A cyclic coloring of a plane graph is a vertex coloring such that vertices incident with the same face have distinct colors. The minimum number of colors in a cyclic coloring of a graph is its cyclic chromatic number χc. Let Δ*be the maximum face degree of a graph. There exist plane graphs with χc= ⌊3/2 Δ*⌋. Ore and Plummer [5] proved that χc≤ 2, Δ*, which bound was improved to ⌊9/5, Δ*⌋ by Borodin, Sanders, and Zhao [1], and to ⌈5/3,Δ*⌉ by Sanders and Zhao [7]. We introduce a new parameter k*, which is the maximum number of vertices that two faces of a graph can have in common, and prove that χc≤ max {Δ*+ 3,k*+ 2, Δ*+ 14, 3, k*+ 6, 18}, and if Δ*≥ 4 and k*≥ 4, then χc≤ Δ*+ 3,k*+ 2. © 2006 Wiley Periodicals, Inc. J Graph Theory

We consider the existence of several different kinds of factors in 4‐connected claw‐free graphs. This is motivated by the following two conjectures which are in fact equivalent by a recent result of the third author. Conjecture 1 (Thomassen): Every 4‐connected line graph is hamiltonian, i.e., has a connected 2‐factor. Conjecture 2 (Matthews and Sumner): Every 4‐connected claw‐free graph is hamiltonian. We first show that Conjecture 2 is true within the class of hourglass‐free graphs, i.e., graphs that do not contain an induced subgraph isomorphic to two triangles meeting in exactly one vertex. Next we show that a weaker form of Conjecture 2 is true, in which the conclusion is replaced by the conclusion that there exists a connected spanning subgraph in which each vertex has degree two or four. Finally we show that Conjectures 1 and 2 are equivalent to seemingly weaker conjectures in which the conclusion is replaced by the conclusion that there exists a spanning subgraph consisting of a bounded number of paths © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 125–136, 2001

For a graphG, letp(G) andc(G) denote the length of a longest path and cycle, respectively. Let p(t,n) be the minimum ofp(G), whereG ranges over allt-tough connected graphs onn vertices. Similarly, let ?(t,n) be the minimum ofc(G), whereG ranges over allt-tough 2-connected graphs onn vertices. It is shown that for fixedt>0 there exist constantsA, B such that p(t,n)=A·log(n) and ?(t,n)·log(?(t,n))=B·log(n). Examples are presented showing that fort=1 there exist constantsA', B' such that p(t,n)=A'·log(n) and ?(t,n)=B'· log(n). It is conjectured that ?(t,n) =B?·log(n) for some constantB?. This conjecture is shown to be valid within the class of 3-connected graphs and, as conjectured in Bondy [1] forl=3, within the class of 2-connectedK1.l-free graphs, wherel is fixed.

A well‐known conjecture in hamiltonian graph theory states that every 2‐tough graph is hamiltonian. We give some equivalent conjectures, e.g., the conjecture that every 2‐tough graph is hamiltonian‐connected.

Purpose: This paper aims to contribute to the formulation of a theory of consciousness based only on computational processes. In this manner, sound computational explanations of qualia and the "hard problem" of consciousness are provided in response to a lack of physical, chemical and psychological explanations. Design/methodology/approach: The study analyses the little that can be objectively known about qualia, and proposes a process that imitates the same effects. Then it applies the process to a robot (using a thought experiment) to understand whether this would produce the same sensations as humans experience. Findings: A computational explanation of qualia and the "hard problem" of consciousness is possible through computational processes. Research limitations/implications: This is a proposal, subject to argumentation and proof. It is a falsifiable theory, meaning that it is possible to test or reject it, as its computational basis allows for a future implementation. Practical implications: Subjective feeling emerges as an evolutionary by-product when there are no strong evolutionary pressures on the brain. Qualia do not involve magic. These aspects of consciousness in robots and in organisations are capable of being manufactured; one can choose whether to build robots and organisations with qualia and subjective experience. Originality/value: To the best of the author's knowledge, no other computational interpretation of these aspects of consciousness exists. However, it is compatible with the multiple draft model of Dennett (1991) and the attention schema theory of Webb and Graziano (2015). [ABSTRACT FROM AUTHOR]

In the m-peripatetic salesman problem (m-PSP), the aim is to determine m edge disjoint Hamiltonian cycles of minimum total cost on a graph. This article introduces new valid inequalities and [...]