  The aim of this paper is to investigate $m$--isometric composition operators
on directed graphs with one circuit. We establish a characterization of
$m$--isometries and prove that complete hyperexpansiveness coincides with
$2$--isometricity within this class. We discuss the $m$--isometric completion
problem for unilateral weighted shifts and for composition operators on
directed graphs with one circuit. The paper is concluded with an affirmative
solution of the Cauchy dual subnormality problem in the subclass with circuit
containing one element.
