On ℳ-contractibility

A1-contractibility and topological contractibility

These notes are an extended transcript of two lectures given at the University of Ottawa during the workshop “Group actions, generalized cohomology theories, and affine algebraic geometry." The first lecture was a general introduction to A1-homotopy theory focusing on motivating the choice of definitions in the construction, together with a definition of A1-contractible spaces. The second lectu...

Contractibility and NP-completeness

For a fixed graph H, let H-CON denote the problem of determining whether a given graph is contractible to H. The complexity of H-CON is studied for H belonging to certain classes of graphs, together covering all connected graphs of order at most 4. In particular, H-CON is NP-complete if H is a connected triangle-free graph other than a star. For each connected graph H of order a t most 4 other ...

Contractibility for open global constraints

Open forms of global constraints allow the addition of new variables to an argument during the execution of a constraint program. Such forms are needed for difficult constraint programming problems where problem construction and problem solving are interleaved, and fit naturally within constraint logic programming. However, in general, filtering that is sound for a global constraint can be unso...

Contractibility of overlapping decentralized control

Contractibility and the Hadwiger Conjecture

Consider the following relaxation of the Hadwiger Conjecture: For each t there exists Nt such that every graph with no Kt minor admits a vertex partition into dαt + βe parts, such that each component of the subgraph induced by each part has at most Nt vertices. The Hadwiger Conjecture corresponds to the case α = 1, β = −1 and Nt = 1. Kawarabayashi and Mohar [K. Kawarabayashi, B. Mohar, A relaxe...