Graph cobordisms and Heegaard Floer homology

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 and connected 3-manifolds, and allows for cobordisms with disconnected ends. As an application, 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 give examples where it is non-trivial on the hat flavor.