On the (parameterized) complexity of recognizing well-covered (r,l)-graphs.

An (r,ℓ)(r,ℓ)-partition of a graph G is a partition of its vertex set into r independent sets and ℓℓ cliques. A graph is (r,ℓ)(r,ℓ) if it admits an (r,ℓ)(r,ℓ)-partition. A graph is well-covered if every maximal independent set is also maximum. A graph is (r,ℓ)(r,ℓ)-well-covered if it is both (r,ℓ)(r,ℓ) and well-covered. In this paper we consider two different decision problems. In the (r,ℓ)(r,ℓ)-Well-Covered Graph problem ((r,ℓ)(r,ℓ) wcg for short), we are given a graph G, and the question is whether G is an (r,ℓ)(r,ℓ)-well-covered graph. In the Well-Covered (r,ℓ)(r,ℓ)-Graph problem (wc (r,ℓ)(r,ℓ) g for short), we are given an (r,ℓ)(r,ℓ)-graph G together with an (r,ℓ)(r,ℓ)-partition of V(G) into r independent sets and ℓℓ cliques, and the question is whether G is well-covered. We classify most of these problems into P, coNP-complete, NP-complete, NP-hard, or coNP-hard. Only the cases wc(r, 0)g for r≥3r≥3 remain open. In addition, we consider the parameterized complexity of these problems for several choices of parameters, such as the size αα of a maximum independent set of the input graph, its neighborhood diversity, or the number ℓℓ of cliques in an (r,ℓ)(r,ℓ)-partition. In particular, we show that the parameterized problem of deciding whether a general graph is well-covered parameterized by αα can be reduced to the wc (0,ℓ)(0,ℓ) g problem parameterized by ℓℓ, and we prove that this latter problem is in XP but does not admit polynomial kernels unless coNP⊆NP/polycoNP⊆NP/poly

Preconscious Prediction of a Driver's Decision Using Intracranial Recordings

Abstract While driving, we make numerous conscious decisions such as route and turn direction selection. Although drivers are held responsible, the neural processes that govern such decisions are not clear. We recorded intracranial EEG signals from six patients engaged in a computer-based driving simulator. Patients decided which way to turn (left/right) and subsequently reported the time of the decision. We show that power modulations of gamma band oscillations (30–100 Hz) preceding the reported time of decision (up to 5.5 sec) allow prediction of decision content with high accuracy (up to 82.4%) on a trial-by-trial basis, irrespective of subsequent motor output. Moreover, these modulations exhibited a spatiotemporal gradient, differentiating left/right decisions earliest in premotor cortices and later in more anterior and lateral regions. Our results suggest a preconscious role for the premotor cortices in early stages of decision-making, which permits foreseeing and perhaps modifying the content of real-life human choices before they are consciously made.</jats:p