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 having the 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 given graph. For this model, we achieve local property reconstructors for the properties of connectivity and $k$-connectivity in undirected graphs, and the property of strong connectivity in directed graphs. Along the way, we present a method of transforming a local reconstructor (which acts as a "adjacency matrix oracle" for the corrected graph) into an "adjacency list oracle". This allows us to recursively use our local reconstructor for $(k-1)$-connectivity to obtain a local reconstructor for $k$-connectivity. We also extend this notion of local property reconstruction to parametrized graph properties (for instance, having diameter at most $D$ for some parameter $D$) and require that the corrected graph has the property with parameter close to 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 corrected graph has diameter roughly 2D. We also exploit a connection between local property reconstruction and property testing, observed by Brakerski, to obtain new tolerant property testers for all of the aforementioned properties. Except for the one for connectivity, these are the first tolerant property testers for these properties.