Circuits or edge-disjoint unions of circuits in a connected graph

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Evaluate the number of nonempty circuits or edge-disjoint unions of circuits in a connected graph
Hi,
I've a question related to the graph theory.

Take a connected graph with ##n## nodes and ##b## edges. We know there are ##m = b - n + 1## fundamental circuits.

Which is the total number of nonempty circuits or edge-disjoint unions of circuits ? If we do not take in account the circuit orientation I believe the answer is ##2^m - 1##.

Is the above correct ? Thanks.
 
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To be more specific consider the following graph with 2 nodes and 4 edges. We have 3 fundamental circuits, but which is the total number of circuits or edge-disjoint unions of circuits ?
appunti.jpg
 

What is a circuit in a connected graph?

A circuit in a connected graph is a closed path that starts and ends at the same vertex, with no repeated edges or vertices along the way.

What is an edge-disjoint union of circuits in a connected graph?

An edge-disjoint union of circuits in a connected graph is a collection of circuits that share no common edges, but together cover all the edges in the graph.

Why are circuits and edge-disjoint unions of circuits important in graph theory?

Circuits and edge-disjoint unions of circuits are important in graph theory because they help us understand the connectivity and structure of a graph. They also have practical applications in network design and optimization problems.

How can we find circuits or edge-disjoint unions of circuits in a connected graph?

There are various algorithms for finding circuits or edge-disjoint unions of circuits in a connected graph, such as the depth-first search algorithm or the Ford-Fulkerson algorithm.

What is the relationship between circuits and edge-disjoint unions of circuits in a connected graph?

Circuits and edge-disjoint unions of circuits are related in that a single circuit can be seen as a special case of an edge-disjoint union of circuits, where there is only one circuit in the union. Additionally, edge-disjoint unions of circuits can be used to decompose a graph into smaller subgraphs, each of which contains a circuit.

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