### 4.3.4.8. four parameters/two final graphs

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

Proposition 145

Proof 143 The quantity $max\left(2,\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}\right)+max\left(2,\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}\right)+max\left(2,\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}\right)-2$ corresponds to the minimum number of variables needed for building two non-empty connected components of respective size $\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}$ and $\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}$. If this quantity is greater than the total number of variables we have that ${\mathrm{\pi \pi \pi }}_{1}\beta €1$.

Proposition 146

$\begin{array}{cc}& \mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\beta §\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }:\hfill \\ & max\left(1,\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}\right)+max\left(1,\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}\right)+max\left(1,\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}\right)>\hfill \\ & {\mathrm{\pi \pi \pi \pi \pi \pi \pi }}_{\mathrm{\pi Έ\pi ½\pi Έ\pi \pi Έ\pi °\pi »}}\beta {\mathrm{\pi \pi \pi }}_{1}\beta €1\hfill \end{array}$

Proof 144 The quantity $max\left(1,\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}\right)+max\left(1,\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}\right)+max\left(1,\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}\right)$ corresponds to the minimum number of variables needed for building two non-empty connected components of respective size $\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}$ and $\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}$. If this quantity is greater than the total number of variables we have that ${\mathrm{\pi \pi \pi }}_{1}\beta €1$.

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

Proposition 147

Proof 145 Similar to PropositionΒ 145.

Proposition 148

$\begin{array}{cc}& \mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\beta §\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }:\hfill \\ & max\left(1,\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}\right)+max\left(1,\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}\right)+max\left(1,\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}\right)>\hfill \\ & {\mathrm{\pi \pi \pi \pi \pi \pi \pi }}_{\mathrm{\pi Έ\pi ½\pi Έ\pi \pi Έ\pi °\pi »}}\beta {\mathrm{\pi \pi \pi }}_{2}\beta €1\hfill \end{array}$

Proof 146 Similar to PropositionΒ 146.

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

Proposition 149

$\begin{array}{cc}& \mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\beta §\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }:\hfill \\ & \mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}\beta \left[β\frac{{\mathrm{\pi \pi \pi \pi \pi \pi \pi }}_{2}}{2}β+1,\hfill \\ & {\mathrm{\pi \pi \pi \pi \pi \pi \pi }}_{\mathrm{\pi Έ\pi ½\pi Έ\pi \pi Έ\pi °\pi »}}-\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}-\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}-1\right]\hfill \end{array}$

Proof 147 First, note that, when ${\mathrm{\pi \pi \pi }}_{2}>1$, we have that $\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}\beta €β\frac{{\mathrm{\pi \pi \pi \pi \pi \pi \pi }}_{2}}{2}β$. Second, note that, when ${\mathrm{\pi \pi \pi }}_{2}\beta €1$, we have that $\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}\beta ₯{\mathrm{\pi \pi \pi \pi \pi \pi \pi }}_{\mathrm{\pi Έ\pi ½\pi Έ\pi \pi Έ\pi °\pi »}}-\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}-\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}$. Since ${\mathrm{\pi \pi \pi }}_{2}$ has to have at least one value the result follows.

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

Proposition 150

$\begin{array}{cc}& \mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\beta §\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }:\hfill \\ & \mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}\beta \left[β\frac{{\mathrm{\pi \pi \pi \pi \pi \pi \pi }}_{1}}{2}β+1,\hfill \\ & {\mathrm{\pi \pi \pi \pi \pi \pi \pi }}_{\mathrm{\pi Έ\pi ½\pi Έ\pi \pi Έ\pi °\pi »}}-\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}-\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}-1\right]\hfill \end{array}$

Proof 148 Similar to PropositionΒ 149.