  We construct maps on hat Heegaard Floer homology for cobordisms decorated
with graphs. The graph TQFT allows for cobordisms with disconnected ends. Our
construction uses Juh\'{a}sz's sutured Floer TQFT. We compute the maps for
several elementary graph cobordisms. As an application, we compute the action
of the fundamental group on hat Heegaard Floer homology.
