  The problem of finding the largest connected subgraph of a given undirected
host graph, subject to constraints on the maximum degree $\Delta$ and the
diameter $D$, was introduced in \cite{maxddbs}, as a generalization of the
Degree-Diameter Problem. A case of special interest is when the host graph is a
common parallel architecture. Here we discuss the case when the host graph is a
$k$-dimensional mesh. We provide some general bounds for the order of the
largest subgraph in arbitrary dimension $k$, and for the particular cases of
$k=3, \Delta = 4$ and $k=2, \Delta = 3$, we give constructions that result in
sharper lower bounds.
