  Exploring the graph approach, we restate the extended definition of
noncontextuality provided by the contextuality-by-default framework. This
extended definition avoids the assumption of nondisturbance, which states that
whenever two contexts overlap, the marginal distribution obtained for the
intersection must be the same. We show how standard tools for characterizing
contextuality can also be used in this extended framework for any set of
measurements and, in addition, we also provide several conditions that can be
tested directly in any contextuality experiment. Our conditions reduce to
traditional ones for noncontextuality if the nondisturbance assumption is
satisfied.
