In this paper, we work with simple and finite graphs. We study a
generalization of the \emph{Cage Problem}, which has been widely studied since
cages were introduced by Tutte \cite{T47} in 1947 and after Erd\" os and Sachs
\cite{ES63} proved their existence in 1963. An \emph{(r,g)-graph} is an
r-regular graph in which the shortest cycle has length equal to g; that is,
it is an r-regular graph with girth g. An \emph{(r,g)-cage} is an
(r,g)-graph with the smallest possible number of vertices among all
(r,g)-graphs; the order of an (r,g)-cage is denoted by n(r,g). The Cage
Problem consists of finding (r,g)-cages; it is well-known that (r,g)-cages
have been determined only for very limited sets of parameter pairs (r,g).
There exists a simple lower bound for n(r,g), given by Moore and denoted by
n0​(r,g). The cages that attain this bound are called \emph{Moore cages}.Comment: 18 page