Among the uncertainties of stellar evolution theory, we investigate how the $^{12}$C($\alpha, \gamma$)$^{16}$O reaction rate affects the evolution of massive stars for the initial masses of $M ({\rm ZAMS})=$ 13 - 40 M$_\odot$ and the solar metallicity. We show that the {\sl explodability} of these stars, i.e., which of a neutron star (NS) or a black hole (BH) is formed, is sensitive to the strength of convective shell burning of C and O, and thus the mass fractions of C ($X$(C)) and O in the shell. For the small $^{12}$C($\alpha, \gamma$)$^{16}$O reaction rate that yields larger $X$(C), $X$(C) is further enhanced by mixing of C from the overlying layer and then C shell burning is strengthened. The extra heating by C shell burning tends to prevent the contraction of outer layers and decrease the {\sl compactness parameter} at $M_r$ = 2.5 M$_\odot$. This effect leads to the formation of smaller mass cores of Si and Fe and steeper density and pressure gradients at the O burning shell in the presupernova models. If the pressure gradient there is steeper, the model is more likely to explode to form a NS rather than a BH. We describe the pressure gradient against $M_r$ with $V/U$ and the density drop with $1/U$, where $U$ and $V$ are non-dimensional variables to describe the stellar structure. We estimate the critical values of $V/U$ and $1/U$ at the O-burning shell above which the model is more likely to explode. We conclude that the smaller $^{12}$C($\alpha, \gamma$)$^{16}$O reaction rate makes the mass range of $M ({\rm ZAMS})$ that forms a NS larger.
Comment: 46 pages, 50 figures