  We consider a special case of $X$-secure $T$-private information retrieval
(XSTPIR), where the security requirement is \emph{asymmetric} due to possible
missing communication links between the $N$ databases considered in the system.
We define the problem with a communication matrix that indicates all possible
communications among the databases, and propose a database grouping mechanism
that collects subsets of databases in an optimal manner, followed by a
group-based PIR scheme to perform asymmetric XSTPIR with the goal of maximizing
the communication rate (minimizing the download cost). We provide an upper
bound on the general achievable rate of asymmetric XSTPIR, and show that the
proposed scheme achieves this upper bound in some cases. The proposed approach
outperforms classical XSTPIR under certain conditions, and the results of this
work show that unlike in the symmetric case, some databases with certain
properties can be dropped to achieve higher rates, concluding that more
databases is not always better.
