This paper studies the algebraic structure of a new class of hyperplane arrangement \(\mathscr {A}\) obtained by deleting two hyperplanes from a free arrangement. We provide some information on the minimal free resolutions of the logarithmic derivation module of \(\mathscr {A}\) , which can be used to compute a lower bound for the graded Betti numbers of the resolution. Specifically, for the three-dimensional case, we determine the minimal free resolution of the logarithmic derivation module of \(\mathscr {A}\) . We present illustrative examples of our main theorems to provide insights into the relationship between algebraic and combinatorial properties for close to free arrangements.
