  We study the complexity of fundamental distributed graph problems in the
recently popular setting where information about the input graph is available
to the nodes before the start of the computation. We focus on the most common
such setting, known as the Supported LOCAL model, where the input graph (on
which the studied graph problem has to be solved) is guaranteed to be a
subgraph of the underlying communication network.
  Building on a successful lower bound technique for the LOCAL model called
round elimination, we develop a framework for proving complexity lower bounds
in the stronger Supported LOCAL model. Our framework reduces the task of
proving a (deterministic or randomized) lower bound for a given problem $\Pi$
to the graph-theoretic task of proving non-existence of a solution to another
problem $\Pi'$ (on a suitable graph) that can be derived from $\Pi$ in a
mechanical manner.
  We use the developed framework to obtain substantial (and, in the majority of
cases, asymptotically tight) Supported LOCAL lower bounds for a variety of
fundamental graph problems, including maximal matching, maximal independent
set, ruling sets, arbdefective colorings, and generalizations thereof. In a
nutshell, for essentially any major lower bound proved in the LOCAL model in
recent years, we prove a similar lower bound in the Supported LOCAL model.
  Our framework also gives rise to a new deterministic version of round
elimination in the LOCAL model: while, previous to our work, the general round
elimination technique required the use of randomness (even for obtaining
deterministic lower bounds), our framework allows to obtain deterministic (and
therefore via known lifting techniques also randomized) lower bounds in a
purely deterministic manner. Previously, such a purely deterministic
application of round elimination was only known for the specific problem of
sinkless orientation [SOSA'23].
