Minimal free resolution of monoimal ideals by iterated mapping cone

In this paper we study minimal free resolutions of some classes of monomial ideals. we first give a sufficient condition to check the minimality of the resolution obtained by the mapping cone. Using it, we obtain the Betti numbers of max-path ideals of rooted trees and ideals containing powers of variables. In particular, we discuss about resolutions of ideals of the form $J_{\mathcal{H}}+(x_{i_1}^2,\ldots, x_{i_m}^2)$ where $J_{\mathcal{H}}$ is the edge ideal of a hypergraph $\mathcal{H}$.