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In this paper, we study the parabolic BGG categories for graded Liesuperalgebras of Cartan type over complex numbers. The gradation of such a Liesuperalgebra $\ggg$ naturally arises, with the zero component $\ggg_0$ being areductive Lie algebra. We first show that there are only two proper parabolicsubalgebras containing Levi subalgebra $\ggg_0$: the ``maximal one" $\sfp_\max$and the ``minimal one" $\sfp_\min$. Furthermore, the parabolic BGG categoryarising from $\sfp_\max$, essentially turns out to be a subcategory of the onearising from $\sfp_\min$. Such a priority of $\sfp_\min$ in the sense ofrepresentation theory reduces the question to the study of the ``minimalparabolic" BGG category $\comi$ associated with $\sfp_\min$. We prove theexistence of projective covers of simple objects in these categories, whichenables us to establish a satisfactory block theory. Most notably, our mainresults are as follows:  (1) We classify and obtain a precise description of the blocks of $\comi$.  (2) We investigate indecomposable tilting and indecomposable projectivemodules in $\comi$, and compute their character formulas.

Parabolic BGG categories and their block decomposition for Lie superalgebras of Cartan type