  Virtual $3$-manifolds were introduced by S.V. Matveev in 2009 as natural
generalizations of the classical $3$-manifolds. In this paper, we introduce a
notion of complexity of a virtual $3$-manifold. We investigate the values of
the complexity for virtual 3-manifolds presented by special polyhedra with one
or two $2$-components. On the basis of these results, we establish the exact
values of the complexity for a wide class of hyperbolic $3$-manifolds with
totally geodesic boundary.

Geometric Topology