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
 

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

What is a circuit in the context of graph theory?

A circuit, also known as a cycle, in graph theory is a closed path where the starting and ending vertices are the same, and no edges or vertices are repeated except for the starting and ending vertex. It represents a sequence of edges that connects a set of vertices without retracing steps.

What does it mean for circuits to be edge-disjoint in a connected graph?

Edge-disjoint circuits in a connected graph refer to circuits that do not share any edges. This means that the edges used in one circuit are completely separate from those used in another circuit, ensuring that each circuit can be traversed without interference from others.

How can we determine the number of edge-disjoint circuits in a connected graph?

The number of edge-disjoint circuits in a connected graph can often be determined using techniques such as the circuit rank or the use of graph connectivity properties. Specifically, one can apply Euler's theorem or utilize algorithms that find maximum edge-disjoint cycles, such as the Edmonds-Karp algorithm or other flow-based methods.

What is the significance of studying edge-disjoint unions of circuits?

Studying edge-disjoint unions of circuits is significant in various applications, including network design, optimization problems, and the study of electrical circuits. Understanding how circuits can be combined without overlap aids in resource allocation, fault tolerance, and improving the efficiency of networks.

Can every connected graph be decomposed into edge-disjoint circuits?

No, not every connected graph can be decomposed into edge-disjoint circuits. The ability to do so depends on the structure of the graph and its properties, such as the presence of cut-edges or bridges. Some graphs may require additional edges or vertices to achieve such a decomposition.

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