Given a directed acyclic graph (DAG) , we say that is -depth-robust (resp. -edge-depth-robust) if for any set (resp. ) of at most nodes (resp. edges) the graph contains a directed path of length . While edge-depth-robust graphs are potentially easier to construct many applications in cryptography require node depth-robust graphs with small indegree. We create a graph reduction that transforms an -edge-depth-robust graph with edges into a -depth-robust graph with nodes and constant indegree. One immediate consequence of this result is the first construction of a provably -depth-robust graph with constant indegree, where previous constructions for had . Our reduction crucially relies on ST-Robust graphs, a new graph property we introduce which may be of independent interest. We say that a directed, acyclic graph with inputs and outputs is -ST-Robust if we can remove any nodes and there exists a subgraph containing at least inputs and outputs such that each of the inputs is connected to all of the outputs. If the graph if -ST-Robust for all we say that the graph is maximally ST-robust. We show how to construct maximally ST-robust graphs with constant indegree and nodes. Given a family of ST-robust graphs and an arbitrary -edge-depth-robust graph we construct a new constant-indegree graph by replacing each node in with an ST-robust graph from . We also show that ST-robust graphs can be used to construct (tight) proofs-of-space and (asymptotically) improved wide-block labeling functions.
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