### 4.3.4.7. three parameters/two final graphs

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

Proposition 135

Proof 133 The quantity $max\left(2,\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}\right)+max\left(3,\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}+1,\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}\right)+max\left(2,\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}\right)-2$ 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 $\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}$ such that $\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}$ is strictly greater than $\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}$. If this quantity is greater than the total number of variables we have that $\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}=\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}$.

Proposition 136

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

Proof 134 The quantity $max\left(1,\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}\right)+max\left(2,\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}+1,\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}\right)+max\left(1,\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}\right)$ 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 $\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}$ such that $\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}$ is strictly greater than $\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}$. If this quantity is greater than the total number of variables we have that $\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}=\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}$.

Proposition 137

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

Proof 135 A value $v$ is not a possible number of vertices for the smallest connected component of type 2 if the following two conditions hold:

• $v+\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}$ does not allow to cover all the vertices of the initial graph: we need at least one extra connected component of type 1 or 2.

• If we add an additional connected component of type 1 or 2 we exceed the number of vertices of the initial graph.

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

Proposition 138

Proof 136 Similar to PropositionΒ 135.

Proposition 139

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

Proof 137 Similar to PropositionΒ 136.

Proposition 140

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

Proof 138 Similar to PropositionΒ 137.

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

Proposition 141

$\begin{array}{cc}& \mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }:\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}=\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}\beta §\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}\mathrm{mod}2=0\beta \hfill \\ & {\mathrm{\pi \pi \pi \pi \pi \pi \pi }}_{2}\mathrm{mod}2={\mathrm{\pi \pi \pi \pi \pi \pi \pi }}_{\mathrm{\pi Έ\pi ½\pi Έ\pi \pi Έ\pi °\pi »}}\mathrm{mod}2\hfill \end{array}$

Proof 139 If the number of vertices of the first graph is even then the number of vertices of the second graph has the same parity as the number of vertices of the initial graph (since a vertex of the initial graph belongs either to the first graph, either to the second graph (but not to both).

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

Proposition 142

$\begin{array}{cc}& \mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }:\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}=\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}\beta §\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}\mathrm{mod}2=0\beta \hfill \\ & {\mathrm{\pi \pi \pi \pi \pi \pi \pi }}_{1}\mathrm{mod}2={\mathrm{\pi \pi \pi \pi \pi \pi \pi }}_{\mathrm{\pi Έ\pi ½\pi Έ\pi \pi Έ\pi °\pi »}}\mathrm{mod}2\hfill \end{array}$

Proof 140 Similar to PropositionΒ 141.

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

Proposition 143

Proof 141 When $\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}+{\mathrm{\pi \pi \pi \pi }}_{2}={\mathrm{\pi \pi \pi \pi \pi \pi \pi }}_{\mathrm{\pi Έ\pi ½\pi Έ\pi \pi Έ\pi °\pi »}}$ there is no more room for an extra connected component for the first final graph.

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

Proposition 144

Proof 142 Similar to PropositionΒ 143.