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POsplitting of a graph into cycles and then into paths
 

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  1. POsplitting of a graph into cycles and then into paths
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    <p>First of all, I should say I am not familiar with the Graph theory and also my mathematics knowledge is very poor. Anyhow I am using graph concepts for my analysis.</p> <p>Basically, I am decomposing an undirected graph (say G) into cycles (closed graph). The specialty of my cycle is that they are the shortest cycles that one can traverse between two vertices (as they are cycle, starting and ending are same though). According to my example graph, my cycles are (1,4,5,1)(1,2,3,4,1)(7,9,8,7) (I neglect the cycles whose length is less than 3). </p> <p>Edit: I use depth first search to get the cycles and then got the smallest cycles.</p> <p>Later, I am further braking those cycles into directed paths. In here, I broke the cycles through the edges (through red lines in figure), so that I inserted starting and ending nodes for my new path graphs. So for the cycle (7,9,8,7)=> new directed paths are (a,9,c)(d,8,7,b) Edit: the further breaking is done only for selected cycles. It is just inserting a new vector and updating the elements. Any graph theory related algorithms doesn't involve here.</p> <p>Then I do some analysis with my data.</p> <blockquote> <p>I did all above things. So, my problem is how to describe the entire things with mathematical notations (without example like I said). This is very hard for me as I do not have even basics.</p> </blockquote> <p>I was trying and googling but still cannot find a way to describe what I did. I guess, the thing what I did is clear for you.</p> <blockquote> <blockquote> <blockquote> <p>So, Could you please help me, How to describe </p> <ol> <li>decomposing a undirected graph into cycles (shortest cycles)</li> <li>Cycle breaking via edges and make directed path graphs (as shown in figure)</li> </ol> </blockquote> </blockquote> <p>with mathematical notation (according to graph theory)</p> </blockquote> <p>I have seen many authors use different notations and symbols to define graphs and their sub graphs, but for me, I can not define such things as my basic are too poor. So, Please help me to say these things in a formal, mathematical way. Thanks in advance.</p> <p>I have inserted sample figures to get idea also. <img src="https://i.stack.imgur.com/PV6mp.jpg" alt="enter image description here"></p> <p>Note: I have add c++ tag as many computer scientists use graph theories and would like to have a response.</p>
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    1. COYou will perhaps receive a better answer @ http://math.stackexchange.com/
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      1. USFredrik Pihl
    2. COthanks for the comments. I posted the same in math.stackexchange and no responses yet...
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      1. USniro
    3. COThat does not make it on-topic here though.
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