In this paper we treat null-controllability properties for the linear Kuramoto-Sivashinsky equation on a network with two types of boundary conditions. More precisely, the equation is considered on a star-shaped tree with Dirichlet and Neumann boundary conditions. By using the moment theory we can derive null-controllability properties with boundary controls acting on the external vertices of the tree. In particular, the controllability holds if the anti-diffusion parameter of the equation does not belong to a critical countable set of real numbers. We point out that the critical set for which the null-controllability fails differs from the first model to the second one.