Dynamic process on graph

Question

I have a question, which I failed to find an answer to myself. I'm a casual user of Mathematica, but do have some knowledge about Dynamic module and other "Advanced" features.

My problem is formulated as follows:

1) I have a graph, which is built on 2*n vertices and contains k (for simplicity we can start with k = 2) perfect matchings on it.

2) I have an operation, that changes the graph by deleting l (again, let us take the simplest case of l = 2) edges from some matching and replacing them with l edges on same set of 2*l vertices.

3) I want to observe a process of random application of such operations to initially supplied graph. And I want to collect some data on every step to further visualize it.

Issues:

Is it possible, using the dynamic module system to perform step-by-step iteration over the data structure? Say I don't want to get the result of application of z random operations on the graph at once, but would rather observe every step with control over the flow by some sort of "next" button.

The issue, as I see it, lies in following: even if I create a dynamic module for my case, and z will be used as a dynamic variable, specifying how many random operation I want to perform, changing it from z to z + 1 will reapply all z + 1 operations, rather than perform a single additional one.

Is there a way to use some kind of history for Dynamic module? Saying if I go forward, use previous result and update it by one?

Hope I've explained it understandably =)

Sergey.

Answer 1

Well, I haven't quite understood your request for a Dynamic approach (I don't see the why or how of it) but since no one else has responded, here is what I can cook up in a few minutes:

Clear[G]
G[n_] := G[n] = 
   Module[{e1, e2, Gr = G[n - 1], m = Length[G[n - 1]], t},
    e1 = RandomInteger[{1, m}];
    e2 = RandomInteger[{1, m}];
    While[Gr[[e1, 2]] != Gr[[e2, 2]] || e1 == e2,
     e2 = RandomInteger[{1, m}];
     ];
    t = Gr[[e1, 1, 2]];
    Gr[[e1, 1, 2]] = Gr[[e2, 1, 2]];
    Gr[[e2, 1, 2]] = t;
    Return[Gr];
    ];
RandomMatching[n_] := Module[{m = {}, i, j, v = Range[1, n]},
  While[Length[v] != 0,
   i = RandomInteger[{1, Length[v]}];
   j = i;
   While[i == j, j = RandomInteger[{1, Length[v]}]];
   i = v[[i]];
   j = v[[j]];
   m = Append[m, i <-> j];
   v = Complement[v, {i, j}];
   ];
  Return[m];
  ]
G[0] = Join[
   Map[Style[#, Red] &, RandomMatching[10]],
   Map[Style[#, Blue] &, RandomMatching[10]],
   Map[Style[#, Green] &, RandomMatching[10]]
   ];

Manipulate[
 Graph[Range[1, 10],
  G[n],
  VertexCoordinates -> 
   Table[{Cos[\[Pi] k/5], Sin[\[Pi] k/5]}, {k, 0, 9}],
  VertexLabels -> "Name",
  ImagePadding -> 40
  ],
 {n, 0, 20, 1}]

I think it's relatively straightforward code, but of course that's just what I think, so if any part of it is confusing, feel free to ask.