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We construct a graph TQFT for the minus flavor of Heegaard Floer homology.Our graph TQFT extends Ozsv\'{a}th and Szab\'{o}'s TQFT for closed andconnected 3-manifolds, and allows for cobordisms with disconnected ends. As anapplication, we give an explicit formula for the chain homotopy type of the$\pi_1$-action on Heegaard Floer homology. We show that on homology the$\pi_1$-action is trivial on the plus, minus and infinity flavors, but giveexamples where it is non-trivial on the hat flavor.

Graph cobordisms and Heegaard Floer homology