  The effects of a boundary on the circuit complexity are studied in two
dimensional theories. The analysis is performed in the holographic realization
of a conformal field theory with a boundary by employing different proposals
for the dual of the complexity, including the "Complexity = Volume" (CV) and
"Complexity = Action" (CA) prescriptions, and in the harmonic chain with
Dirichlet boundary conditions. In all the cases considered except for CA, the
boundary introduces a subleading logarithmic divergence in the expansion of the
complexity as the UV cutoff vanishes. Holographic subregion complexity is also
explored in the CV case, finding that it can change discontinuously under
continuous variations of the configuration of the subregion.
