  Spin models that have been proposed to describe dimerized chains, ladders,
two dimensional antiferromagnets, and other compounds are here studied when
some spins are replaced by spinless vacancies, such as it occurs by $Zn$
doping. A small percentage of vacancies rapidly destroys the spin gap, and
their presence induces enhanced antiferromagnetic correlations near those
vacancies. The study is performed with computational techniques which includes
Lanczos, world-line Monte Carlo, and the Density Matrix Renormalization Group
methods. Since the phenomenon of enhanced antiferromagnetism is found to occur
in several models and cluster geometries, a common simple explanation for its
presence may exist. It is argued that the resonating-valence-bond character of
the spin correlations at short distances of a large variety of models is
responsible for the presence of robust staggered spin correlations near
vacancies and lattice edges. The phenomenon takes place regardless of the long
distance properties of the ground state, and it is caused by a ``pruning'' of
the available spin singlets in the vicinity of the vacancies. The effect
produces a broadening of the low temperature NMR signal for the compounds
analyzed here. This broadening should be experimentally observable in the
structurally dimerized chain systems
  $Cu(NO_3)_2\cdot2.5H_2O$, $CuWO_4$, $(VO)_2P_2O_7$, and
$Sr_{14}Cu_{24}O_{41}$, in ladder materials such as $Sr Cu_2 O_3$, in the
spin-Peierls systems $CuGeO_3$ and $NaV_2 O_5$, and in several others since it
is a universal effect common to a wide variety of models and compounds.
