### 4.3.4.8. four parameters/two final graphs

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

Proposition 145

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

Proof 143 The quantity $max\left(2,\mathrm{𝐌𝐈𝐍}_{\mathrm{𝐍𝐂𝐂}}_{1}\right)+max\left(2,\mathrm{𝐌𝐀𝐗}_{\mathrm{𝐍𝐂𝐂}}_{1}\right)+max\left(2,\mathrm{𝐌𝐈𝐍}_{\mathrm{𝐍𝐂𝐂}}_{2}\right)-2$ corresponds to the minimum number of variables needed for building two non-empty connected components of respective size $\mathrm{𝐌𝐈𝐍}_{\mathrm{𝐍𝐂𝐂}}_{1}$ and $\mathrm{𝐌𝐀𝐗}_{\mathrm{𝐍𝐂𝐂}}_{1}$. If this quantity is greater than the total number of variables we have that ${\mathrm{𝐍𝐂𝐂}}_{1}\le 1$.

Proposition 146

$\begin{array}{cc}& \mathrm{𝚟𝚙𝚊𝚛𝚝𝚒𝚝𝚒𝚘𝚗}\wedge \mathrm{𝚌𝚘𝚗𝚜𝚎𝚌𝚞𝚝𝚒𝚟𝚎}_\mathrm{𝚕𝚘𝚘𝚙𝚜}_\mathrm{𝚊𝚛𝚎}_\mathrm{𝚌𝚘𝚗𝚗𝚎𝚌𝚝𝚎𝚍}:\hfill \\ & max\left(1,\mathrm{𝐌𝐈𝐍}_{\mathrm{𝐍𝐂𝐂}}_{1}\right)+max\left(1,\mathrm{𝐌𝐀𝐗}_{\mathrm{𝐍𝐂𝐂}}_{1}\right)+max\left(1,\mathrm{𝐌𝐈𝐍}_{\mathrm{𝐍𝐂𝐂}}_{2}\right)>\hfill \\ & {\mathrm{𝐍𝐕𝐄𝐑𝐓𝐄𝐗}}_{\mathrm{𝙸𝙽𝙸𝚃𝙸𝙰𝙻}}⇒{\mathrm{𝐍𝐂𝐂}}_{1}\le 1\hfill \end{array}$

Proof 144 The quantity $max\left(1,\mathrm{𝐌𝐈𝐍}_{\mathrm{𝐍𝐂𝐂}}_{1}\right)+max\left(1,\mathrm{𝐌𝐀𝐗}_{\mathrm{𝐍𝐂𝐂}}_{1}\right)+max\left(1,\mathrm{𝐌𝐈𝐍}_{\mathrm{𝐍𝐂𝐂}}_{2}\right)$ corresponds to the minimum number of variables needed for building two non-empty connected components of respective size $\mathrm{𝐌𝐈𝐍}_{\mathrm{𝐍𝐂𝐂}}_{1}$ and $\mathrm{𝐌𝐀𝐗}_{\mathrm{𝐍𝐂𝐂}}_{1}$. If this quantity is greater than the total number of variables we have that ${\mathrm{𝐍𝐂𝐂}}_{1}\le 1$.

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

Proposition 147

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

Proof 145 Similar to Proposition 145.

Proposition 148

$\begin{array}{cc}& \mathrm{𝚟𝚙𝚊𝚛𝚝𝚒𝚝𝚒𝚘𝚗}\wedge \mathrm{𝚌𝚘𝚗𝚜𝚎𝚌𝚞𝚝𝚒𝚟𝚎}_\mathrm{𝚕𝚘𝚘𝚙𝚜}_\mathrm{𝚊𝚛𝚎}_\mathrm{𝚌𝚘𝚗𝚗𝚎𝚌𝚝𝚎𝚍}:\hfill \\ & max\left(1,\mathrm{𝐌𝐈𝐍}_{\mathrm{𝐍𝐂𝐂}}_{2}\right)+max\left(1,\mathrm{𝐌𝐀𝐗}_{\mathrm{𝐍𝐂𝐂}}_{2}\right)+max\left(1,\mathrm{𝐌𝐈𝐍}_{\mathrm{𝐍𝐂𝐂}}_{1}\right)>\hfill \\ & {\mathrm{𝐍𝐕𝐄𝐑𝐓𝐄𝐗}}_{\mathrm{𝙸𝙽𝙸𝚃𝙸𝙰𝙻}}⇒{\mathrm{𝐍𝐂𝐂}}_{2}\le 1\hfill \end{array}$

Proof 146 Similar to Proposition 146.

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

Proposition 149

$\begin{array}{cc}& \mathrm{𝚟𝚙𝚊𝚛𝚝𝚒𝚝𝚒𝚘𝚗}\wedge \mathrm{𝚌𝚘𝚗𝚜𝚎𝚌𝚞𝚝𝚒𝚟𝚎}_\mathrm{𝚕𝚘𝚘𝚙𝚜}_\mathrm{𝚊𝚛𝚎}_\mathrm{𝚌𝚘𝚗𝚗𝚎𝚌𝚝𝚎𝚍}:\hfill \\ & \mathrm{𝐌𝐈𝐍}_{\mathrm{𝐍𝐂𝐂}}_{2}\notin \left[⌊\frac{{\mathrm{𝐍𝐕𝐄𝐑𝐓𝐄𝐗}}_{2}}{2}⌋+1,\hfill \\ & {\mathrm{𝐍𝐕𝐄𝐑𝐓𝐄𝐗}}_{\mathrm{𝙸𝙽𝙸𝚃𝙸𝙰𝙻}}-\mathrm{𝐌𝐈𝐍}_{\mathrm{𝐍𝐂𝐂}}_{1}-\mathrm{𝐌𝐀𝐗}_{\mathrm{𝐍𝐂𝐂}}_{1}-1\right]\hfill \end{array}$

Proof 147 First, note that, when ${\mathrm{𝐍𝐂𝐂}}_{2}>1$, we have that $\mathrm{𝐌𝐈𝐍}_{\mathrm{𝐍𝐂𝐂}}_{2}\le ⌊\frac{{\mathrm{𝐍𝐕𝐄𝐑𝐓𝐄𝐗}}_{2}}{2}⌋$. Second, note that, when ${\mathrm{𝐍𝐂𝐂}}_{2}\le 1$, we have that $\mathrm{𝐌𝐈𝐍}_{\mathrm{𝐍𝐂𝐂}}_{2}\ge {\mathrm{𝐍𝐕𝐄𝐑𝐓𝐄𝐗}}_{\mathrm{𝙸𝙽𝙸𝚃𝙸𝙰𝙻}}-\mathrm{𝐌𝐈𝐍}_{\mathrm{𝐍𝐂𝐂}}_{1}-\mathrm{𝐌𝐀𝐗}_{\mathrm{𝐍𝐂𝐂}}_{1}$. Since ${\mathrm{𝐍𝐂𝐂}}_{2}$ has to have at least one value the result follows.

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

Proposition 150

$\begin{array}{cc}& \mathrm{𝚟𝚙𝚊𝚛𝚝𝚒𝚝𝚒𝚘𝚗}\wedge \mathrm{𝚌𝚘𝚗𝚜𝚎𝚌𝚞𝚝𝚒𝚟𝚎}_\mathrm{𝚕𝚘𝚘𝚙𝚜}_\mathrm{𝚊𝚛𝚎}_\mathrm{𝚌𝚘𝚗𝚗𝚎𝚌𝚝𝚎𝚍}:\hfill \\ & \mathrm{𝐌𝐈𝐍}_{\mathrm{𝐍𝐂𝐂}}_{1}\notin \left[⌊\frac{{\mathrm{𝐍𝐕𝐄𝐑𝐓𝐄𝐗}}_{1}}{2}⌋+1,\hfill \\ & {\mathrm{𝐍𝐕𝐄𝐑𝐓𝐄𝐗}}_{\mathrm{𝙸𝙽𝙸𝚃𝙸𝙰𝙻}}-\mathrm{𝐌𝐈𝐍}_{\mathrm{𝐍𝐂𝐂}}_{2}-\mathrm{𝐌𝐀𝐗}_{\mathrm{𝐍𝐂𝐂}}_{2}-1\right]\hfill \end{array}$

Proof 148 Similar to Proposition 149.