The conjugacy problem in three types of group constructions involving cyclic subgroups is discussed. First it is shown that if G has the solvable conjugacy problem and if
h
∈
G
h \in G
and
k
∈
G
k \in G
satisfy (a) h and k are not power conjugate to themselves or each other, (b) the power conjugacy problem in G with respect to h or k is solvable, and (c) the double coset solvability problem in G is solvable with respect to
⟨
h
⟩
\langle h\rangle
and
⟨
k
⟩
\langle k\rangle
, then the HNN extension
G
∗
=
⟨
G
,
t
;
t
−
1
h
t
=
k
⟩
{G^ \ast } = \langle G,t;{t^{ - 1}}ht = k\rangle
has the solvable conjugacy problem. This result is used to deduce a similar theorem for free products with amalgamation, a fact first stated by Lipschutz. Then it is shown that if A and B are groups with the solvable conjugacy problem and
h
∈
A
h \in A
and
k
∈
B
k \in B
taken with themselves satisfy the conditions above in A and B, respectively, then
⟨
A
∗
B
;
[
h
,
k
]
=
1
⟩
\langle A ^\ast B;[h,k] = 1\rangle
has the solvable conjugacy problem.