We fabricate and study quantum dot structures incorporating quasi-one-dimensional excited states. The structures are realized by graded bandgap GaAs/AlGaAs quantum wires self-formed inside inverted tetrahedral pyramids. The ground state transitions exhibit typical characteristics of fully confined excitons, including single photon emission. Efficient carrier thermalization and relaxation, as well as correlated photon emission is observed also among the excited states, indicating the formation of quasi-one-dimensional multi-exciton states. These structures offer interesting possibilities for collecting and directing charge carriers towards heterostructured potential traps. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4721660]