A New Connection Between Node and Edge Depth Robust Graphs

We create a graph reduction that transforms an $(e, d)$-edge-depth-robust graph with $m$ edges into a $(e/4,d)$-depth-robust graph with $O(m)$ nodes and constant indegree. An $(e,d)$-depth robust graph is a directed, acyclic graph with the property that that after removing any $e$ nodes of the graph there remains a path with length at least $d$. Similarly, an $(e, d)$-edge-depth robust graph is a directed, acyclic graph with the property that after removing any $e$ edges of the graph there remains a path with length at least $d$. Our reduction relies on constructing graphs with a property we define and analyze called ST-Robustness. We say that a directed, acyclic graph with $n$ inputs and $n$ outputs is $(k_1, k_2)$-ST-Robust if we can remove any $k_1$ nodes and there exists a subgraph containing at least $k_2$ inputs and $k_2$ outputs such that each of the $k_2$ inputs is connected to all of the $k_2$ outputs. We use our reduction on a well known edge-depth-robust graph to construct an $(\frac{n \log \log n}{\log n}, \frac{n}{\log n (\log n)^{\log \log n}})$-depth-robust graph.