  We study a class of Labelled Oriented Graph (LOG) group where the underlying
graph is a tadpole graph. We show that such a group is the natural HNN
extension of a cyclically presented group and investigate the relationship
between the LOG group and the cyclically presented group. We relate the second
homotopy groups of their presentations and show that hyperbolicity of the
cyclically presented group implies solvability of the conjugacy problem for the
LOG group. In the case where the label on the tail of the LOG spells a positive
word in the vertices in the circuit we show that the LOGs and groups coincide
with those considered by Szczepanski and Vesnin. We obtain new presentations
for these cyclically presented groups and show that the groups of Fibonacci
type introduced by Johnson and Mawdesley are of this form. These groups
generalize the Fibonacci groups and the Sieradski groups and have been studied
by various authors. We continue these investigations, using small cancellation
and curvature methods to obtain results on hyperbolicity, automaticity,
SQ-universality, and solvability of decision problems.
