Distance bounds for graphs with some negative Bakry-\'Emery curvature

We prove distance bounds for graphs possessing positive Bakry-'Emery curvature apart from an exceptional set, where the curvature is allowed to be non-positive. If the set of non-positively curved vertices is finite, then the graph admits an explicit upper bound for the diameter. Otherwise, the graph is a subset of the tubular neighborhood with an explicit radius around the non-positively curved vertices. Those results seem to be the first assuming non-constant Bakry-'Emery curvature assumptions on graphs.