EXAMPLE: As a first example consider the group of invariants denoted by Proposition 68, which relate the number of arcs $\mathrm{𝐍𝐀𝐑𝐂}$ with the number of vertices of the smallest and largest connected component (i.e., $\mathrm{𝐌𝐈𝐍}\_\mathrm{𝐍𝐂𝐂}$ and $\mathrm{𝐌𝐀𝐗}\_\mathrm{𝐍𝐂𝐂}$).

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 $\mathrm{𝐌𝐈𝐍}\_{\mathrm{𝐍𝐂𝐂}}^2+\mathrm{𝐌𝐀𝐗}\_{\mathrm{𝐍𝐂𝐂}}^2$ is greater than or equal to $\mathrm{𝐌𝐈𝐍}\_\mathrm{𝐍𝐂𝐂}+\mathrm{𝐌𝐀𝐗}\_\mathrm{𝐍𝐂𝐂}-2+(\mathrm{𝐌𝐈𝐍}\_\mathrm{𝐍𝐂𝐂}=1))$, 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 Proposition 51, which relate the number of arcs $\mathrm{𝐍𝐀𝐑𝐂}$ to the number of vertices $\mathrm{𝐍𝐕𝐄𝐑𝐓𝐄𝐗}$ according to the arc generator (see Figure 2.3.4) used for generating the initial digraph: