Circuit rank

In graph theory, a branch of mathematics, the circuit rank of an undirected graph is the minimum number of edges that must be removed from the graph to break all its cycles, making it into a tree or forest. Alternatively, it can be interpreted as the number of independent cycles in the graph. Unlike the corresponding feedback arc set problem for directed graphs, the circuit rank r is easily computed using the formula

where m is the number of edges in the given graph, n is the number of vertices, and c is the number of connected components. It is also possible to construct a minimum-size set of edges that breaks all cycles efficiently, either using a greedy algorithm or by complementing a spanning forest.

The circuit rank is also known as the cyclomatic number or nullity of the graph. It can be explained in terms of algebraic graph theory as the dimension of the cycle space of a graph, in terms of matroid theory using the corank of a graphic matroid, and in terms of topology as one of the Betti numbers of a topological space derived from the graph. It counts the ears in an ear decomposition of the graph, forms the basis of parameterized complexity on almost-trees, and has been applied in software metrics as part of the definition of cyclomatic complexity of a piece of code. Under the name of cyclomatic number, the concept was introduced by Gustav Kirchhoff.

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