Graph decomposition and union by mathematical notation

Question

(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.

Answer 1

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.