Switching with more than two colours

AbstractThe operation of switching a finite graph was introduced by Seidel, in the study of strongly regular graphs. We may conveniently regard a graph as being a 2-colouring of a complete graph; then the extension to switching of an m-coloured complete graph is easy to define. However, the situation is very different. For m>2, all m-coloured graphs lie in the same switching class. However, there are still interesting things to say, especially in the infinite case.This paper presents the basic theory of switching with more than two colours. In the finite case, all graphs on a given set of vertices are equivalent under switching, and we determine the structure of the switching group and show that its extension by the symmetric group on the vertex set is primitive.In the infinite case, there is more than one switching class; we determine all those for which the group of switching automorphisms is the symmetric group. We also exhibit some other cases (including the random m-coloured complete graph) where the group of switching-automorphisms is highly transitive.Finally we consider briefly the case where not all switchings are allowed. For convenience, we suppose that there are three colours of which two may be switched. We show that, in the case of almost all finite random graphs, the analogue of the bijection between switching classes and two-graphs holds

Complementation, Local Complementation, and Switching in Binary Matroids

In 2004, Ehrenfeucht, Harju, and Rozenberg showed that any graph on a vertex set $V$ can be obtained from a complete graph on $V$ via a sequence of the operations of complementation, switching edges and non-edges at a vertex, and local complementation. The last operation involves taking the complement in the neighbourhood of a vertex. In this paper, we consider natural generalizations of these operations for binary matroids and explore their behaviour. We characterize all binary matroids obtainable from the binary projective geometry of rank $r$ under the operations of complementation and switching. Moreover, we show that not all binary matroids of rank at most $r$ can be obtained from a projective geometry of rank $r$ via a sequence of the three generalized operations. We introduce a fourth operation and show that, with this additional operation, we are able to obtain all binary matroids.Comment: Fixed an error in the proof of Theorem 5.3. Adv. in Appl. Math. (2020

Symmetric Strategy Improvement

Symmetry is inherent in the definition of most of the two-player zero-sum games, including parity, mean-payoff, and discounted-payoff games. It is therefore quite surprising that no symmetric analysis techniques for these games exist. We develop a novel symmetric strategy improvement algorithm where, in each iteration, the strategies of both players are improved simultaneously. We show that symmetric strategy improvement defies Friedmann's traps, which shook the belief in the potential of classic strategy improvement to be polynomial

Domain-independent neural underpinning of task-switching: an fMRI investigation

The ability to shift between different tasks according to internal or external demands, which is at the core of our behavioral flexibility, has been generally linked to the functionality of left fronto-parietal regions. Traditionally, the left and right hemispheres have also been associated with verbal and spatial processing, respectively. We therefore investigated with functional MRI whether the processes engaged during task-switching interact in the brain with the domain of the tasks to be switched, that is, verbal or spatial. Importantly, physical stimuli were exactly the same and participants\u2019 performance was matched between the two domains. The fMRI results showed a clearly left-lateralized involvement of fronto-parietal regions when contrasting task-switching vs. single task blocks in the context of verbal rules. A more bilateral pattern, especially in the prefrontal cortex, was instead observed for switching between spatial tasks. Moreover, while a conjunction analysis showed that the core regions involved in task-switching, independently of the switching context, were localized both in left inferior prefrontal and parietal cortices and in bilateral supplementary motor area, a direct analysis of functional lateralization revealed that hemispheric asymmetries in the frontal lobes were more biased toward the left side for the verbal domain than for the spatial one and vice versa. Overall, these findings highlight the role of left fronto-parietal regions in task-switching, above and beyond the specific task requirements, but also show that hemispheric asymmetries may be modulated by the more specific nature of the tasks to be performed during task-switching