Opposed to the other graph attributes, registers are available for all graph objects, not just for represented graphs. If a graph is represented, manipulations of its skeleton ensure that the register attributes are updated accordingly.

- Orderings of the node set (e.g. st-numberings)
- Partitions of the node set (e.g. clique partitions)

- Orderings of the arc set (e.g. Euler cycles)
- Partitions of the arc set (e.g. partitions into 1-matchings)
- Implicit arc orientations (e.g. feedback arc sets) Other than the subgraph multiplicities, edge colours are stored by plain arrays. So, especially for geometric graph instances, the number of arcs is large relative to the size of memory representation. It is therefore recommended to handle subgraph incidence vectors by edge colours only if non-trivial lower capacity bounds prevent from using the subgraph multiplicities.

Any directed path encoded into the predecessor labels can be backtracked from its end node without scanning the incidence lists of the intermediate nodes. So, whenever possible, subgraphs are stored by the predecessor register:

- Simple directed paths and cycles
- Rooted trees
- One-cycle trees consisting of a directed cycle and some trees pointing away from this cycle
- Any node disjoint union of the listed subgraph types