### 4.3.4.10. six parameters/two final graphs

$\mathrm{𝐌𝐀𝐗}_{\mathrm{𝐍𝐂𝐂}}_{1}$, $\mathrm{𝐌𝐀𝐗}_{\mathrm{𝐍𝐂𝐂}}_{2}$, $\mathrm{𝐌𝐈𝐍}_{\mathrm{𝐍𝐂𝐂}}_{1}$, $\mathrm{𝐌𝐈𝐍}_{\mathrm{𝐍𝐂𝐂}}_{2}$, ${\mathrm{𝐍𝐂𝐂}}_{1}$, ${\mathrm{𝐍𝐂𝐂}}_{2}$

Proposition 165

$\begin{array}{cc}& \mathrm{𝚊𝚙𝚊𝚛𝚝𝚒𝚝𝚒𝚘𝚗}\wedge \mathrm{𝐚𝐫𝐜}_\mathrm{𝐠𝐞𝐧}=\mathrm{𝑃𝐴𝑇𝐻}\wedge {\mathrm{𝐍𝐕𝐄𝐑𝐓𝐄𝐗}}_{\mathrm{𝙸𝙽𝙸𝚃𝙸𝙰𝙻}}>0:\hfill \\ & \alpha ·\mathrm{𝐌𝐈𝐍}_{\mathrm{𝐍𝐂𝐂}}_{1}+\mathrm{𝐌𝐀𝐗}_{\mathrm{𝐍𝐂𝐂}}_{1}+\hfill \\ & \beta ·\mathrm{𝐌𝐈𝐍}_{\mathrm{𝐍𝐂𝐂}}_{2}+\mathrm{𝐌𝐀𝐗}_{\mathrm{𝐍𝐂𝐂}}_{2}\le {\mathrm{𝐍𝐕𝐄𝐑𝐓𝐄𝐗}}_{\mathrm{𝙸𝙽𝙸𝚃𝙸𝙰𝙻}}+{\mathrm{𝐍𝐂𝐂}}_{1}+{\mathrm{𝐍𝐂𝐂}}_{2}-1,\mathrm{𝚠𝚒𝚝𝚑}:\hfill \\ & \left\{\begin{array}{c}•\alpha =max\left(0,{\mathrm{𝐍𝐂𝐂}}_{1}-1\right),\hfill \\ •\beta =max\left(0,{\mathrm{𝐍𝐂𝐂}}_{2}-1\right).\hfill \end{array}\right\\hfill \end{array}$

Proof 163 Let $CC\left({G}_{1}\right)=\left\{C{C}_{a}^{1}:a\in \left[\mathrm{𝐍𝐂𝐂}1\right]\right\}$ and $CC\left({G}_{2}\right)=\left\{C{C}_{a}^{2}:a\in \left[\mathrm{𝐍𝐂𝐂}2\right]\right\}$ 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 ${\left({A}_{i}\right)}_{i\in \left[{\mathrm{𝐍𝐂𝐂}}_{1}+{\mathrm{𝐍𝐂𝐂}}_{2}\right]}$ and there exists $j\in \left[2\right]$ such that ${A}_{i}\in CC\left({G}_{1+\left(j\mathrm{mod}2\right)}\right)$, for $i\mathrm{mod}2=0$ and ${A}_{i}\in CC\left({G}_{1+\left(\left(j+1\right)\mathrm{mod}2\right)}\right)$ for $i\mathrm{mod}2=1$ and ${A}_{i}\cap {A}_{i+1}\ne \varnothing$ for $i\in \left[{\mathrm{𝐍𝐂𝐂}}_{1}+{\mathrm{𝐍𝐂𝐂}}_{2}-1\right]$.

By inclusion-exclusion principle, since ${A}_{i}\cap {A}_{j}=\varnothing$ whenever $j\ne i+1$, we obtain ${\mathrm{𝐍𝐕𝐄𝐑𝐓𝐄𝐗}}_{\mathrm{𝙸𝙽𝙸𝚃𝙸𝙰𝙻}}={\Sigma }_{a\in \left[\mathrm{𝐍𝐂𝐂}1\right]}|C{C}_{a}^{1}|+{\Sigma }_{a\in \left[\mathrm{𝐍𝐂𝐂}2\right]}|C{C}_{a}^{2}|-{\Sigma }_{i\in \left[\mathrm{𝐍𝐂𝐂}1+\mathrm{𝐍𝐂𝐂}2-1\right]}|{A}_{i}\cap {A}_{i+1}|$. Since $|{A}_{i}\cap {A}_{i+1}|$ is equal to 1 for every well defined $i$, we obtain ${\Sigma }_{a\in \left[\mathrm{𝐍𝐂𝐂}1\right]}|C{C}_{a}^{1}|+{\Sigma }_{a\in \left[\mathrm{𝐍𝐂𝐂}2\right]}|C{C}_{a}^{2}|={\mathrm{𝐍𝐕𝐄𝐑𝐓𝐄𝐗}}_{\mathrm{𝙸𝙽𝙸𝚃𝙸𝙰𝙻}}+\mathrm{𝐍𝐂𝐂}1+\mathrm{𝐍𝐂𝐂}2-1$.

Since $\alpha ·\mathrm{𝐌𝐈𝐍}_{\mathrm{𝐍𝐂𝐂}}_{1}+\mathrm{𝐌𝐀𝐗}_{\mathrm{𝐍𝐂𝐂}}_{1}+\beta ·\mathrm{𝐌𝐈𝐍}_{\mathrm{𝐍𝐂𝐂}}_{2}+\mathrm{𝐌𝐀𝐗}_{\mathrm{𝐍𝐂𝐂}}_{2}\le {\Sigma }_{a\in \left[\mathrm{𝐍𝐂𝐂}1\right]}|C{C}_{a}^{1}|+{\Sigma }_{a\in \left[\mathrm{𝐍𝐂𝐂}2\right]}|C{C}_{a}^{2}|$ the result follows.

Proposition 166

$\begin{array}{cc}& \mathrm{𝚊𝚙𝚊𝚛𝚝𝚒𝚝𝚒𝚘𝚗}\wedge \mathrm{𝐚𝐫𝐜}_\mathrm{𝐠𝐞𝐧}=\mathrm{𝑃𝐴𝑇𝐻}\wedge {\mathrm{𝐍𝐕𝐄𝐑𝐓𝐄𝐗}}_{\mathrm{𝙸𝙽𝙸𝚃𝙸𝙰𝙻}}>0:\hfill \\ & \alpha ·\mathrm{𝐌𝐀𝐗}_{\mathrm{𝐍𝐂𝐂}}_{1}+\mathrm{𝐌𝐈𝐍}_{\mathrm{𝐍𝐂𝐂}}_{1}+\hfill \\ & \beta ·\mathrm{𝐌𝐀𝐗}_{\mathrm{𝐍𝐂𝐂}}_{2}+\mathrm{𝐌𝐈𝐍}_{\mathrm{𝐍𝐂𝐂}}_{2}\ge {\mathrm{𝐍𝐕𝐄𝐑𝐓𝐄𝐗}}_{\mathrm{𝙸𝙽𝙸𝚃𝙸𝙰𝙻}}+{\mathrm{𝐍𝐂𝐂}}_{1}+{\mathrm{𝐍𝐂𝐂}}_{2}-1,\mathrm{𝚠𝚒𝚝𝚑}:\hfill \\ & \left\{\begin{array}{c}•\alpha =max\left(0,{\mathrm{𝐍𝐂𝐂}}_{1}-1\right),\hfill \\ •\beta =max\left(0,{\mathrm{𝐍𝐂𝐂}}_{2}-1\right).\hfill \end{array}\right\\hfill \end{array}$

Proof 164 Similar to Proposition 165.