  In this work a systematic approach to the search for all-$sp^2$ bonded carbon
allotropes with low density is presented. In particular, we obtain a number of
novel energetically stable crystal structures, whose arrangement is closely
related to the topology of graphene, by modifying the packing of congruent
discs under the condition of local stability. Our procedure starts from an
initial parent topology and proceeds to generate daughter architectures derived
by lowering the packing factors. Furthermore, we assess both the electronic
properties, such as the band structure and the density of states, and the
mechanical properties, such as the elastic constants and the stress--strain
characteristics, of parent's and daughter's geometries from first-principle
simulations. We find, using geometrical packing arguments, that some
arrangements lead to a density as low as half that of graphene, obtaining some
of the least dense structures of all-$sp^2$ bonded carbon allotropes that could
ever be synthesized. Nevertheless, a threshold value of the density exists
below which the mechanical rigidity of graphene is irreparably lost, while
keeping other mechanical characteristics, such as the specific toughness and
strength, almost unchanged with lower weight.
