When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ITCS.2021.64
URN: urn:nbn:de:0030-drops-136038
URL: https://drops.dagstuhl.de/opus/volltexte/2021/13603/
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### A New Connection Between Node and Edge Depth Robust Graphs

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### Abstract

Given a directed acyclic graph (DAG) G = (V,E), we say that G is (e,d)-depth-robust (resp. (e,d)-edge-depth-robust) if for any set S ⊂ V (resp. S ⊆ E) of at most |S| ≤ e nodes (resp. edges) the graph G-S contains a directed path of length d. 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 (e, d)-edge-depth-robust graph with m edges into a (e/2,d)-depth-robust graph with O(m) nodes and constant indegree. One immediate consequence of this result is the first construction of a provably ((n log log n)/log n, n/{(log n)^{1 + log log n}})-depth-robust graph with constant indegree, where previous constructions for e = (n log log n)/log n had d = O(n^{1-ε}). 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 n inputs and n outputs is (k₁, k₂)-ST-Robust if we can remove any k₁ nodes and there exists a subgraph containing at least k₂ inputs and k₂ outputs such that each of the k₂ inputs is connected to all of the k₂ outputs. If the graph if (k₁,n-k₁)-ST-Robust for all k₁ ≤ n we say that the graph is maximally ST-robust. We show how to construct maximally ST-robust graphs with constant indegree and O(n) nodes. Given a family 𝕄 of ST-robust graphs and an arbitrary (e, d)-edge-depth-robust graph G we construct a new constant-indegree graph Reduce(G, 𝕄) by replacing each node in G with an ST-robust graph from 𝕄. We also show that ST-robust graphs can be used to construct (tight) proofs-of-space and (asymptotically) improved wide-block labeling functions.

### BibTeX - Entry

```@InProceedings{blocki_et_al:LIPIcs.ITCS.2021.64,
author =	{Jeremiah Blocki and Mike Cinkoske},
title =	{{A New Connection Between Node and Edge Depth Robust Graphs}},
booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
pages =	{64:1--64:18},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-177-1},
ISSN =	{1868-8969},
year =	{2021},
volume =	{185},
editor =	{James R. Lee},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},