Mincuts from graphs to find permutations that break the system into disjoint sets?

Question

Assume an undirected system, graph where you want to find out elements in their permutations that makes the system disjoint such that each set is minimum. This graph has two special nodes: sink and source that cannot be in the elements.

How can you computationally find the minimum cuts given some graph G=(V,E)?

Answer 1

Feel free to choose any method to calculate mincuts computationally from graphs. I list below a simple example, relevant research, models, storage methods, lemmas, theorems -- and something about visualisation and computing on which this thread is focused on. Next step after the simple example is the parametrisation of the graphical model and then computation.

Simple example with Python and Cartesian product

Assume a 3x2x2 graph with 3 parallel where 1st branch with 3 things, 2nd branch with 2 things and last branch 2 things. The minimum cuts are {{1,4,6},{1,4,7},{1,5,6},{1,5,7},{2,4,6},{2,4,7},{2,5,6},{2,5,7},{3,4,6},{3,4,7},{3,5,6},{3,5,7}} .

import itertools;

somelists = [
   [1, 2, 3],
   [4, 5],
   [6, 7]
]

print list(itertools.product(*somelists))

Computing

• General question How to analyse a sparse adjacency matrix? but relevant because graphs often are sparse.

• Graph Complement useful in finding mincuts: a trial with Mathematica below but notice that some bugs in Mathematica found in at least 10.1 related to mincut and VertexConnectivity commands.

Mathematics

• [Graphical parametrization problem in computational algebra for cuts] Cut ideals such as Betti numbers of cut ideals of trees

• [Hard graphical parametrization problem of simple graphs] Smallest generating set of simple graphs for fixed number of vertices and fixed number of vertex cuts?

• The mincuts have crossing criteria, cactus storage method and the current research outlined in the Graph Connectivity paper (2004) .

• Many relevant terms for minimum cuts such as vertex separators, bifurcators, vertex cuts, edge cuts, graph boundaries -- mathematical terms outlined here .

Visualising series-parallel graphs with sink and source

• Tikz package to draw series parallel graphs with many vars in series parallel

• Quality material contain "Graphical models" by Lauritzen Steffen, "Graphical models for R" and Algebraic geometric material by Sturmfels.