how to write the process of decomposition of a graph into shortest closed sub graphs

Question

If I want to decompose a graph in to possible shortest closed cycles (as shown in right side).

then how can i describe this process with mathematical notations.

to understand please refer below figure. As i have a criteria to get cycles whose length is more than 3, i want to include that criteria also in to the mathematical definition (so that i left cycles whose length is less than 3 in the right side closed graphs)

please, help me to explain this in a formal mathematical way. for example, if i start this notation, it could be looking like

Let G be a undirected graph and decomposable into set of sub graphs S1, S, .. Sk of the edges of G.

Answer 1

If I understand correctly, the question is essentially "how do I phrase this problem?" Firstly, what you have drawn is not a graph decomposition for two reasons:

1. There are edges, such as $34$ absent from all of the subgraphs.

2. There is an edge, $23$ which belong to two subgraphs.

So, here's what I think you're trying to write:

Let $G$ be a graph, and let $G^*$ denote $G$ after deletion of its bridges. Let $\Gamma=\Gamma(G^*)$ denote the set of (edge) cycle covers of $G^*$. For any cycle $S$, let $|V(S)|$ denote the number of vertices in $S$. What is $$\min_{C \in \Gamma} \max_{S \in C} |V(S)|$$ and which cycle cover(s) achieves this minimum?

The above formula minimizes the "worst case" cycle lengths.

Your figure suggests you might be particularly interested in planar graphs. If so, you might be seeking "faces", and not just "cycles". (There will be some additional technical details in this case, e.g., after deleting the bridges, we might end up with isolated vertices.)