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(I guess, readers might misguided by my original post, So I modify it)

If I have an undirected graph, Could you please help me to describe

  1. Decomposing a undirected graph into cycles
  2. Cycle breaking into 2 sections (via any 2 graph edges) and make 2 directed path graphs
  3. Obtainig big cycle by getting union of the cycles through the removal of common sharing edges

with mathematical notation (according to graph theory) Please help me.

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Write each edge as an ordered pair. A graph is a set of ordered pairs.

Your final "greatest cycle" is a "circuit" of all nodes, and visits several nodes twice. There may not be a unique shortest circuit, and there's no known efficient way to find a/the shortest circuit: The problem is a version of the so-called "Postman problem".

If you need more, you need to read up on graph theory. I can't name a good introduction, but if google is not doing it for you, perhaps someone else can answer with a suitable recommendation.

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thank you for the comments. Yes greatest cycle is not a shortest circuit. So, I will update the post. Actually, I got all what I need. I want to describe all those things with mathematical notations (according to graph theory). I want assistant for that. – gnp Mar 1 '13 at 12:12
Could you please simply show me how to describe these decomposition, union and unwanted edge removal with mathematical notations. thanks – gnp Mar 1 '13 at 14:29
Write each edge as an ordered pair. A graph is a set of ordered pairs. The rest you need to do yourself (see updated answer). – alexis Mar 1 '13 at 14:41

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