### 4.3.2. Format of an invariant

As we previously saw, we have graph invariants that hold for any digraph as well as tighter graph invariants for specific graph classes. As a consequence, we partition the database in groups of graph invariants. A group of graph invariants corresponds to several invariants such that all invariants relate the same subset of graph parameters and such that all invariants are variations of the first invariant of the group taking into accounts the graph class. Therefore, the first invariant of a group has no precondition, while all other invariants have a non -empty precondition that characterises the graph class for which they hold.

EXAMPLE: As a first example consider the group of invariants denoted by Proposition 68, which relate the number of arcs $\mathrm{\pi \pi \pi \pi }$ with the number of vertices of the smallest and largest connected component (i.e.,Β $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }$ and $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }$).

On the one hand, since the first rule has no precondition it corresponds to a general graph invariant. On the other hand the second rule specifies a tighter condition (since $\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}^{2}+\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}^{2}$ is greater than or equal to $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }+\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }-2+\left(\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }=1\right)\right)$, which only holds for a final graph that is reflexive, symmetric and transitive.

EXAMPLE: As a second example, consider the following group of invariants corresponding to PropositionΒ 51, which relate the number of arcs $\mathrm{\pi \pi \pi \pi }$ to the number of vertices $\mathrm{\pi \pi \pi \pi \pi \pi \pi }$ according to the arc generator (see FigureΒ 2.2.4) used for generating the initial digraph:

$\mathrm{\pi \pi \pi \pi }\beta €{\mathrm{\pi \pi \pi \pi \pi \pi \pi }}^{2}$