Diagonal complexes for surfaces of finite type and surfaces with involution

Two constructions are studied that are inspired by the ideas of a recent paper by the authors.
— The diagonal complex $\mathcal{D}$ and its barycentric subdivision $\mathcal{BD}$ related to an oriented surface of finite type $F$ equipped with a number of labeled marked points. This time, unlike the paper mentioned above, boundary components without marked points are allowed, called holes.
— The symmetric diagonal complex $\mathcal{D}^{\text{inv}}$ and its barycentric subdivision $\mathcal{BD}^{\text{inv}}$ related to a symmetric (=with an involution) oriented surface $F$ equipped with a number of (symmetrically placed) labeled marked points.
The symmetric complex is shown to be homotopy equivalent to the complex of a surface obtained by “taking a half” of the initial symmetric surface.