  A seminal milestone in lattice statistics is the exact solution of the
enumeration of dimers on a simple-quartic net obtained by Fisher,Kasteleyn, and
Temperley (FKT) in 1961. An outstanding related and yet unsolved problem is the
enumeration of dimers on a net with vacant sites. Here we consider this
vacant-site problem with a single vacancy occurring at certain specific sites
on the boundary of a simple-quartic net. First, using a bijection between dimer
and spanning tree configurations due to Temperley, Kenyon, Propp, and Wilson,
we establish that the dimer generating function is independent of the location
of the vacancy, and deduce a closed-form expression for the generating
function. We next carry out finite-size analyses of this solution as well as
that of the FKT solution. Our analyses lead to a logarithmic correction term in
the large-size expansion for the vacancy problem with free boundary conditions.
A concrete example exhibiting this difference is given. We also find the
central charge c=-2 in the language of conformal field theory for the vacancy
problem, as versus the value c=1 when there is no vacancy.
