  A relative one-relator presentation has the form P = < X,H ; R > where X is a
set, H is a group, and R is a group word on X and H. We show that if the group
word on X obtained from R by deleting all the terms from H has what we call the
unique max-min property, then the group defined by P is residually finite if
and only if H is residually finite (Theorem 1). We apply this to obtain new
results concerning the residual finiteness of (ordinary) one-relator groups
(Theorem 4). We also obtain results concerning the conjugacy problem for
one-relator groups (Theorem 5), and results concerning the relative asphericity
of presentations of the form P (Theorem 6).
