# Finding all Minimal Cut Sets for Site Percolation?

Bug introduced in 9.0 and persisting through 11.3.0
FindVertexCut and FindMinimumCut are new in 9.0.

I want to disconnect the left side from the right side. The minimal cut set $C$ is a set of vertices by which the lattice $\mathbb L-C$ becomes disconnected. Consider the example below $\mathbb L=\mathbb L^2$ where an example minimal cut is $\{63,62\}$ for the below example: neither the set $\{62\}$ nor $\{62\}$ are minimal

SeedRandom;
dimension = 10;
coDimension = 10;
percProbability = 0.489;
deleteMe =
Pick[Table[i, {i, dimension*coDimension}],
Table[RandomReal[] > percProbability, {i,
dimension*coDimension}]];
G = GridGraph[{dimension, coDimension}, VertexLabels -> "Name",
G = SetProperty[G, VertexCoordinates -> GraphEmbedding[G]];
H = VertexDelete[G, deleteMe]
(*ConnectedComponents[H]*)
FindVertexCut[H, 2, 97]
FindVertexCut[H]
FindMinimumCut[H] where neither the FindVertexCut[] or FindMinimumCut[] do not work correctly: they suggest to remove isolated vertices and a compontent or not running, without anything to do with minimal vertex cuts. Because Mathematica's command on cuts not working so

How can I find all minimal cut sets for a site percolation?

• Have you tried anything with FindMinimumCut[]? – dr.blochwave Jul 10 '16 at 11:04
• @blochwave I added a small example where you can see both the FindMinimumCut and FindVertexCut do not work on the grid graph so I would not use the functions. – hhh Jul 10 '16 at 15:56