Given a directed acyclic graph (DAG) G=(V,E), we say that G is (e,d)depthrobust (resp. (e,d)edgedepthrobust) 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 edgedepthrobust graphs are potentially easier to construct, many applications in cryptography require node depthrobust graphs with small indegree. We create a graph reduction that transforms an (e,d)edgedepthrobust graph with m edges into a (e/2,d)depthrobust graph with O(m) nodes and constant indegree. One immediate consequence of this result is the first construction of a provably (nloglognlogn,nlogn(logn)loglogn)depthrobust graph with constant indegree. Our reduction crucially relies on STrobust 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)STrobust 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)STrobust for all k1≤n we say that the graph is maximally STrobust. We show how to construct maximally STrobust graphs with constant indegree and O(n) nodes. Given a family M of STrobust graphs and an arbitrary (e,d)edgedepthrobust graph G we construct a new constantindegree graph Reduce(G,M) by replacing each node in G with an STrobust graph from M. We also show that STrobust graphs can be used to construct (tight) proofsofspace and (asymptotically) improved wideblock labeling functions.
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The effect of adding randomly weighted edges
We consider the following question. We have a dense regular graph $G$ with degree $\a n$, where $\a>0$ is a constant. We add $m=o(n^2)$ random edges. The edges of the augmented graph $G(m)$ are given independent edge weights $X(e),e\in E(G(m))$. We estimate the minimum weight of some specified combinatorial structures. We show that in certain cases, we can obtain the same estimate as is known for the complete graph, but scaled by a factor $\a^{1}$. We consider spanning trees, shortest paths and perfect matchings in (pseudorandom) bipartite graphs.
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 Award ID(s):
 1952285
 NSFPAR ID:
 10320466
 Date Published:
 Journal Name:
 SIAM journal on discrete mathematics
 Volume:
 35
 ISSN:
 10957146
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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