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A local property reconstructor for a graph property is an algorithm which,given oracle access to the adjacency list of a graph that is "close" to havingthe property, provides oracle access to the adjacency matrix of a "correction"of the graph, i.e. a graph which has the property and is close to the givengraph. For this model, we achieve local property reconstructors for theproperties of connectivity and $k$-connectivity in undirected graphs, and theproperty of strong connectivity in directed graphs. Along the way, we present amethod of transforming a local reconstructor (which acts as a "adjacency matrixoracle" for the corrected graph) into an "adjacency list oracle". This allowsus to recursively use our local reconstructor for $(k-1)$-connectivity toobtain a local reconstructor for $k$-connectivity.  We also extend this notion of local property reconstruction to parametrizedgraph properties (for instance, having diameter at most $D$ for some parameter$D$) and require that the corrected graph has the property with parameter closeto the original. We obtain a local reconstructor for the low diameter property,where if the original graph is close to having diameter $D$, then the correctedgraph has diameter roughly 2D.  We also exploit a connection between local property reconstruction andproperty testing, observed by Brakerski, to obtain new tolerant propertytesters for all of the aforementioned properties. Except for the one forconnectivity, these are the first tolerant property testers for theseproperties.

Local reconstructors and tolerant testers for connectivity and diameter