Editing Cyclomatic number (section)

===Ear decomposition===
The cyclomatic number 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 [[k-vertex-connected graph|2-vertex-connected]] if 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 <math>r</math>, every open ear decomposition has exactly <math>r</math> ears.<ref>{{citation
 | last = Whitney | first = H. | authorlink = Hassler Whitney
 | journal = [[Transactions of the American Mathematical Society]]
 | pages = 339–362
 | title = Non-separable and planar graphs
 | volume = 34
 | year = 1932
 | issue = 2 | doi=10.2307/1989545 | jstor = 1989545 | pmid = 16587624 | doi-access = free
 | pmc = 1076008
 }}. See in particular Theorems 18 (relating ear decomposition to circuit rank) and 19 (on the existence of ear decompositions).</ref>