### 4.3.4.10. six parameters/two final graphs

$\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}$, $\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}$, $\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}$, $\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}$, ${\mathrm{\pi \pi \pi }}_{1}$, ${\mathrm{\pi \pi \pi }}_{2}$

Proposition 165

Proof 163 Let $CC\left({G}_{1}\right)=\left\{C{C}_{a}^{1}:a\beta \left[\mathrm{\pi \pi \pi }1\right]\right\}$ and $CC\left({G}_{2}\right)=\left\{C{C}_{a}^{2}:a\beta \left[\mathrm{\pi \pi \pi }2\right]\right\}$ be respectively the set of connected components of the first and the second final graphs. Since the initial graph is a path, and since each arc of the initial graph belongs to the first or to the second final graphs (but not to both), there exists ${\left({A}_{i}\right)}_{i\beta \left[{\mathrm{\pi \pi \pi }}_{1}+{\mathrm{\pi \pi \pi }}_{2}\right]}$ and there exists $j\beta \left[2\right]$ such that ${A}_{i}\beta CC\left({G}_{1+\left(j\mathrm{mod}2\right)}\right)$, for $i\mathrm{mod}2=0$ and ${A}_{i}\beta CC\left({G}_{1+\left(\left(j+1\right)\mathrm{mod}2\right)}\right)$ for $i\mathrm{mod}2=1$ and for $i\beta \left[{\mathrm{\pi \pi \pi }}_{1}+{\mathrm{\pi \pi \pi }}_{2}-1\right]$.

By inclusion-exclusion principle, since ${A}_{i}\beta ©{A}_{j}=\mathrm{\beta  }$ whenever , we obtain ${\mathrm{\pi \pi \pi \pi \pi \pi \pi }}_{\mathrm{\pi Έ\pi ½\pi Έ\pi \pi Έ\pi °\pi »}}={\mathrm{\Xi £}}_{a\beta \left[\mathrm{\pi \pi \pi }1\right]}|C{C}_{a}^{1}|+{\mathrm{\Xi £}}_{a\beta \left[\mathrm{\pi \pi \pi }2\right]}|C{C}_{a}^{2}|-{\mathrm{\Xi £}}_{i\beta \left[\mathrm{\pi \pi \pi }1+\mathrm{\pi \pi \pi }2-1\right]}|{A}_{i}\beta ©{A}_{i+1}|$. Since $|{A}_{i}\beta ©{A}_{i+1}|$ is equal to 1 for every well defined $i$, we obtain ${\mathrm{\Xi £}}_{a\beta \left[\mathrm{\pi \pi \pi }1\right]}|C{C}_{a}^{1}|+{\mathrm{\Xi £}}_{a\beta \left[\mathrm{\pi \pi \pi }2\right]}|C{C}_{a}^{2}|={\mathrm{\pi \pi \pi \pi \pi \pi \pi }}_{\mathrm{\pi Έ\pi ½\pi Έ\pi \pi Έ\pi °\pi »}}+\mathrm{\pi \pi \pi }1+\mathrm{\pi \pi \pi }2-1$.

Since $\mathrm{\Xi ±}Β·\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}+\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}+\mathrm{\Xi ²}Β·\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}+\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}\beta €{\mathrm{\Xi £}}_{a\beta \left[\mathrm{\pi \pi \pi }1\right]}|C{C}_{a}^{1}|+{\mathrm{\Xi £}}_{a\beta \left[\mathrm{\pi \pi \pi }2\right]}|C{C}_{a}^{2}|$ the result follows.

Proposition 166

Proof 164 Similar to PropositionΒ 165.