Proposition 145

Proof 143 The quantity $max(2,\mathrm{𝐌𝐈𝐍}\_{\mathrm{𝐍𝐂𝐂}}_1)+max(2,\mathrm{𝐌𝐀𝐗}\_{\mathrm{𝐍𝐂𝐂}}_1)+max(2,\mathrm{𝐌𝐈𝐍}\_{\mathrm{𝐍𝐂𝐂}}_2)-2$ 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

Proof 144 The quantity $max(1,\mathrm{𝐌𝐈𝐍}\_{\mathrm{𝐍𝐂𝐂}}_1)+max(1,\mathrm{𝐌𝐀𝐗}\_{\mathrm{𝐍𝐂𝐂}}_1)+max(1,\mathrm{𝐌𝐈𝐍}\_{\mathrm{𝐍𝐂𝐂}}_2)$ 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 147

Proof 145 Similar to Proposition 145.

Proposition 148

Proof 146 Similar to Proposition 146.

Proposition 149

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.

Proposition 150

Proof 148 Similar to Proposition 149.