  We show that the main idea behind contextuality-by-default (CbD), i.e., the
assumption that a physical measurement has to be understood as a contextual
collection of random variables, is implicit in the compatibility-hypergraph
approach to contextuality (CA) and use this result to develop in the latter
important concepts which were introduced in the former. We introduce in CA the
non-degeneracy condition, which is the analogous of consistent connectedness,
and prove that this condition is, in general, weaker than non-disturbance
condition. The set of non-degenerate behaviours defines a polytope, implying
that one can characterize consistent connectedness using linear inequalities.
We introduce the idea of extended contextuality for behaviours and prove that a
behaviour is non-contextual in the standard sense iff it is non-degenerate and
non-contextual in the extended sense. Finally, we use extended scenarios and
behaviours to shed new light on our results.
