

Equation: cycle[ dft[ ugraph[] ] ]

Explanation:
checks cycles over an undirected graph using depth-first traversals.


Equation: cycle[ cycle[ dft[ ugraph[] ] ]]

Precondition errors: 
   duplicate copies of a  cycle component.


Equation: cycle[ cycle[ cycle[ dft[ ugraph[] ] ] ] ]

Precondition errors: 
   duplicate copies of a  cycle component.


Equation: strongc[ dft[ ugraph[]]]

Prerestriction errors: 
   a directed graph is expected between ugraph and strongc.


Equation: strongc[ ugraph[]]

Prerestriction errors: 
   a directed graph is expected between ugraph and strongc.


Equation: strongc[ cycle[ bft[ ugraph[]]]]

Prerestriction errors: 
   a depth-first-traversal layer is expected below cycle.
   a directed graph is expected between ugraph and strongc.


Equation: number[ dft[ region[ dft[ dgraph[]]]]]

Precondition errors: 
   region is not expecting a search component (dft) above it.
   dft is not expecting a search component (dft) above it.
   duplicate copies of a  dft component.
Prerestriction errors: 
   an undirected graph layer is expected between dgraph and region.


Equation: strongc[ dft[ region[ dft[ dgraph[]]]]]

Precondition errors: 
   region is not expecting a search component (dft) above it.
   dft is not expecting a search component (dft) above it.
   duplicate copies of a  dft component.
Prerestriction errors: 
   an undirected graph layer is expected between dgraph and region.


Equation: number[ cycle[ region[ dft[ ugraph[]]]]]

Explanation:
numbers vertices, and checks cycles, and finds connected-regions over an undirected graph using depth-first traversals.
