The circuit rank of a graph may be described using matroid theory as the corank of the graphic matroid of. Using the greedy property of matroids, this means that one can find a minimum set of edges that breaks all cycles using a greedy algorithm that at each step chooses an edge that belongs to at least one cycle of the remaining graph. Alternatively, a minimum set of edges that breaks all cycles can be found by constructing a spanning forest of and choosing the complementary set of edges that do not belong to the spanning forest.
The number of independent cycles
In algebraic graph theory, the circuit rank is also the dimension of the cycle space of. Intuitively, this can be explained as meaning that the circuit rank counts the number of independent cycles in the graph, where a collection of cycles is independent if it is not possible to form one of the cycles as the symmetric difference of some subset of the others. This count of independent cycles can also be explained using homology theory, a branch of topology. Any graph may be viewed as an example of a 1-dimensional simplicial complex, a type of topological space formed by representing each graph edge by a line segment and gluing these line segments together at their endpoints. The cyclomatic number is the rank of the first homology group of this complex, Because of this topological connection, the cyclomatic number of a graph is also called the first Betti number of. More generally, the first Betti number of any topological space, defined in the same way, counts the number of independent cycles in the space.
Applications
Meshedness coefficient
A variant of the circuit rank for planar graphs, normalized by dividing by the maximum possible circuit rank of any planar graph with the same vertex set, is called the meshedness coefficient. For a connected planar graph with edges and vertices, the meshedness coefficient can be computed by the formula Here, the numerator of the formula is the circuit rank of the given graph, and the denominator is the largest possible circuit rank of an -vertex planar graph. The meshedness coefficient ranges between 0 for trees and 1 for maximal planar graphs.
Ear decomposition
The circuit rank controls the number of ears in an ear decomposition of a graph, a partition of the edges of the graph into paths and cycles that is useful in many graph algorithms. In particular, a graph is 2-vertex-connectedif and only if it has an open ear decomposition. This is a sequence of subgraphs, where the first subgraph is a simple cycle, the remaining subgraphs are all simple paths, each path starts and ends on vertices that belong to previous subgraphs, and each internal vertex of a path appears for the first time in that path. In any biconnected graph with circuit rank, every open ear decomposition has exactly ears.
Almost-trees
A graph with cyclomatic number is also called a r-almost-tree, because only r edges need to be removed from the graph to make it into a tree or forest. A 1-almost-tree is a near-tree: a connected near-tree is a pseudotree, a cycle with a tree rooted at each vertex. Several authors have studied the parameterized complexity of graph algorithms on r-near-trees, parameterized by.
Generalizations to directed graphs
The cycle rank is an invariant of directed graphs that measures the level of nesting of cycles in the graph. It has a more complicated definition than circuit rank and is more difficult to compute. Another problem for directed graphs related to the circuit rank is the minimum feedback arc set, the smallest set of edges whose removal breaks all directed cycles. Both cycle rank and the minimum feedback arc set are NP-hard to compute. It is also possible to compute a simpler invariant of directed graphs by ignoring the directions of the edges and computing the circuit rank of the underlying undirected graph. This principle forms the basis of the definition of cyclomatic complexity, a software metric for estimating how complicated a piece of computer code is.
In the fields of chemistry and cheminformatics, the circuit rank of a molecular graph is sometimes referred to as the Frèrejacque number.
Related concepts
Other numbers defined in terms of edge deletion from undirected graphs include the edge connectivity, the minimum number of edges to delete in order to disconnect the graph, and matching preclusion, the minimum number of edges to delete in order to prevent the existence of a perfect matching.