  Electron transport through nanodevices of atoms in a single-layer rectangular
arrangement with free (open) boundary conditions parallel to the direction of
the current flow is studied within the single-band tight binding model. The
Landauer formula gives the electrical conductance to be a function of the
electron transmission probability, ${\cal T}(E)$, as a function of the energy
$E$ of the incoming electron. A quantum dragon nanodevice is one which has a
perfectly conducting channel, namely ${\cal T}(E)=1$ for all energies which are
transmitted by the external leads even though there may be arbitrarily strong
electron scattering. The rectangular single-layer systems are shown to be able
to be quantum dragon devices, both for uniform leads and for dimerized leads.
The quantum dragon condition requires appropriate lead-device connections and
correlated randomness in the device.
