  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.
