  We consider the problem of enumerating d-irreducible maps, i.e. planar maps
whose all cycles have length at least d, and such that any cycle of length d is
the boundary of a face of degree d. We develop two approaches in parallel: the
natural approach via substitution where these maps are obtained from general
maps by a replacement of all d-cycles by elementary faces, and a bijective
approach via slice decomposition which consists in cutting the maps along
shortest paths. Both lead to explicit expressions for the generating functions
of d-irreducible maps with controlled face degrees, summarized in some elegant
"pointing formula". We provide an equivalent description of d-irreducible
slices in terms of so-called d-oriented trees. We finally show that irreducible
maps give rise to a hierarchy of discrete integrable equations which include
equations encountered previously in the context of naturally embedded trees.
