  We study domain wall induced by spontaneously broken $\mathbb{Z}_2$ symmetry
and its gravitational wave signature in the standard model with a complex
scalar in connection with dark matter physics. In a minimal setup, a linear
term of the singlet field is added to the scalar potential as an explicit
$\mathbb{Z}_2$ breaking term to make the domain wall unstable. We obtain its
minimal size from cosmological constraints and show that the parameter space
that can be probed by current and future pulsar time array experiments requires
the vacuum expectation value of the singlet field to be greater than
$\mathcal{O}(10-100)$ TeV, along with a singlet-like Higgs mass of
$\mathcal{O}(1-100)$ TeV. However, such a region is severely restricted by the
dark matter relic density, which places an upper bound on the singlet vacuum
expectation value at approximately 200 TeV, and limits the dark matter mass to
about half of the singlet-like Higgs boson mass.
