Compute the minimum cycle basis (MCB) of a graph. A cycle basis is a set of
cycles that can be combined to generate all the cycles of a graph. The MCB is
the basis of minimum weight (length). As an example, there are three cycles
in
naphthalene.
To generate the three cycles (cycle space) we actually only need two of the
three cycles. The third cycle can be generated by combining the other two.
Each combination of the two cycles is called a cycle basis. There is one
basis with the two rings of size six and two bases with either ring of size
six and the perimeter ring of size 10. The weights of each basis are 12
(6+6), 16 (6+10), 16 (6+10). The basis of the two six member rings has
minimum weight (12) and is thus the MCB of
naphthalene.
The Smallest Set of Smallest Rings (SSSR) is normally used interchangeably
with MCB although traditionally their meaning was different (
Berger, F. et. al.. J. Chem. Inf. Comput. Sci.. 2004. 44). Although the MCB can be used to generate all other cycles of a
graph it may not be unique. A compound with a bridged ring system, such as
3-quinuclidinol, has three equally valid minimum cycle bases. From
any two six member rings the third ring can be generated by ⊕-sum
(xor-ing) their edges. As there may be more than one MCB it should not be used
as a
unique descriptor. The smallest (but potentially exponential)
unique set of short cycles which generates the cycle space is the
union of all minimum cycle bases. This set is known as the
RelevantCycles. The intersect of all minimum cycle bases (
EssentialCycles) is also unique. Although the number of
EssentialCycles is polynomial it can not always be used to generate the
cycle space.