  We report a gravitational $BF$-type action principle propagating two
(complex) degrees of freedom that, besides the gauge connection and the $B$
field, only employs an additional Lagrange multiplier. The action depends on
two parameters and remarkably is polynomial in the $B$ field. For a particular
choice of the involved parameters the action provides an alternative
description of (complex) general relativity with a nonvanishing cosmological
constant, whereas another choice corresponds to anti-self-dual gravity. Generic
values of the parameters produce "close neighbors" of general relativity,
although there is a peculiar choice of the parameters that leads to a
Hamiltonian theory with two scalar constraints. Given the nontrivial form of
the resulting scalar constraint for these models, we consider a more general
setting where the scalar constraint is replaced with an arbitrary analytic
function of some fundamental variables and show that the Poisson algebra
involving this constraint together with the Gauss and vector constraints of the
Ashtekar formalism closes, thus generating an infinite family of gravitational
models that propagate the same number of degrees of freedom as general
relativity.
