### 4.3.4.9. five parameters/two final graphs

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

Proposition 151

$\begin{array}{cc}& \mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\beta §\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }:\hfill \\ & \mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}Β·max\left(0,{\mathrm{\pi \pi \pi }}_{1}-1\right)+\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}+\hfill \\ & \mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}Β·max\left(0,{\mathrm{\pi \pi \pi }}_{1}-2\right)+\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}\beta €{\mathrm{\pi \pi \pi \pi \pi \pi \pi }}_{\mathrm{\pi Έ\pi ½\pi Έ\pi \pi Έ\pi °\pi »}}\hfill \end{array}$

Proof 149 The left-hand side ofΒ 151 corresponds to the minimum number of vertices of the two final graphs provided that we build the smallest possible connected components.

Proposition 152

$\begin{array}{cc}& \mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\beta §\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }:\hfill \\ & {\mathrm{\pi \pi \pi }}_{1}\beta €\left(\stackrel{Β―}{\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}}>0\right)+β\frac{\mathrm{\Xi ±}}{\mathrm{\Xi ²}}β+\left(\mathrm{\Xi ±}\mathrm{mod}\mathrm{\Xi ²}\beta ₯max\left(1,\underset{Μ²}{\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}}\right)\right)\hfill \\ & \left\{\begin{array}{c}\beta ’\mathrm{\Xi ±}=max\left(0,{\mathrm{\pi \pi \pi \pi \pi \pi \pi }}_{\mathrm{\pi Έ\pi ½\pi Έ\pi \pi Έ\pi °\pi »}}-max\left(1,\underset{Μ²}{\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}}\right)-max\left(1,\underset{Μ²}{\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}}\right)\right),\hfill \\ \beta ’\mathrm{\Xi ²}=max\left(1,\underset{Μ²}{\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}}\right)+max\left(1,\underset{Μ²}{\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}}\right).\hfill \end{array}\right\\hfill \end{array}$

Proof 150 The maximum number of connected components is achieved by building non-empty groups as small as possible, except for two groups of respective size $max\left(1,\underset{Μ²}{\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}}\right)$ and $max\left(1,\underset{Μ²}{\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}}\right)$, which have to be built.

Proposition 153

$\begin{array}{cc}& \mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\beta §\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }:\hfill \\ & \mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}Β·max\left(0,{\mathrm{\pi \pi \pi }}_{1}-1\right)+\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}+\hfill \\ & \mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}Β·{\mathrm{\pi \pi \pi }}_{1}+\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}\beta ₯{\mathrm{\pi \pi \pi \pi \pi \pi \pi }}_{\mathrm{\pi Έ\pi ½\pi Έ\pi \pi Έ\pi °\pi »}}\hfill \end{array}$

Proof 151 The left-hand side ofΒ 153 corresponds to the maximum number of vertices of the two final graphs provided that we build the largest possible connected components.

Proposition 154

$\begin{array}{cc}& \mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\beta §\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }:\hfill \\ & {\mathrm{\pi \pi \pi }}_{1}\beta ₯\left(\stackrel{Β―}{\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}}<{\mathrm{\pi \pi \pi \pi \pi \pi \pi }}_{\mathrm{\pi Έ\pi ½\pi Έ\pi \pi Έ\pi °\pi »}}\right)+β\frac{\mathrm{\Xi ±}}{\mathrm{\Xi ²}}β+\left(\mathrm{\Xi ±}\mathrm{mod}\mathrm{\Xi ²}>\stackrel{Β―}{\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}}\right)\hfill \\ & \left\{\begin{array}{c}\beta ’\mathrm{\Xi ±}=max\left(0,{\mathrm{\pi \pi \pi \pi \pi \pi \pi }}_{\mathrm{\pi Έ\pi ½\pi Έ\pi \pi Έ\pi °\pi »}}-\stackrel{Β―}{\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}}-\stackrel{Β―}{\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}},\hfill \\ \beta ’\mathrm{\Xi ²}=max\left(1,\stackrel{Β―}{\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}}\right)+max\left(1,\stackrel{Β―}{\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}}\right).\hfill \end{array}\right\\hfill \end{array}$

Proof 152 The minimum number of connected components is achieved by taking the groups as large as possible except for two groups of respective size $\stackrel{Β―}{\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}}$ and $\stackrel{Β―}{\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}}$, which have to be built.

Proposition 155

Proof 153 If ${\mathrm{\pi \pi \pi }}_{1}\beta €1$ we have that $\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}\beta €\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}$. Otherwise, when ${\mathrm{\pi \pi \pi }}_{1}>1$, we have that $\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}Β·max\left(0,{\mathrm{\pi \pi \pi }}_{1}-1\right)+\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}+\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}+\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}+\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}Β·max\left(0,{\mathrm{\pi \pi \pi }}_{1}-3\right)\beta €{\mathrm{\pi \pi \pi \pi \pi \pi \pi }}_{\mathrm{\pi Έ\pi ½\pi Έ\pi \pi Έ\pi °\pi »}}$. ${\mathrm{\pi \pi \pi }}_{1}-3$ comes from the fact that we build the minimum number of connected components in the second final graph (i.e.,Β ${\mathrm{\pi \pi \pi }}_{1}-1$ connected components) and that we have already built two connected components of respective size $\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}$ and $\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}$. By isolating $\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}$ in the previous expression and by grouping the two inequalities the result follows.

Proposition 156

Proof 154 The maximum number of connected components of ${G}_{1}$ is achieved by:

• Building a first connected component of ${G}_{1}$ involving $\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}$ vertices,

• Building a first connected component of ${G}_{2}$ involving $\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}$ vertices,

• Building alternatively a connected component of ${G}_{1}$ and a connected component of ${G}_{2}$ involving respectively $\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}$ and $\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}$ vertices,

• Finally, if this is possible, building a connected component of ${G}_{1}$ involving $\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}$ vertices.

Proposition 157

Proof 155 The minimum number of connected components of ${G}_{1}$ is achieved by:

• Building a first connected component of ${G}_{2}$ involving $\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}$ vertices,

• Building a first connected component of ${G}_{1}$ involving $\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}$ vertices,

• Building alternatively a connected component of ${G}_{2}$ and a connected component of ${G}_{1}$ involving respectively $\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}$ and $\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}$ vertices,

• Finally, if this is possible, building a connected component of ${G}_{2}$ involving $\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}$ vertices and a connected component of ${G}_{1}$ with the remaining vertices. Note that these remaining vertices cannot be incorporated in the connected components previously built.

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

Proposition 158

$\begin{array}{cc}& \mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\beta §\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }:\hfill \\ & \mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}Β·max\left(0,{\mathrm{\pi \pi \pi }}_{2}-1\right)+\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}+\hfill \\ & \mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}Β·max\left(0,{\mathrm{\pi \pi \pi }}_{2}-2\right)+\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}\beta €{\mathrm{\pi \pi \pi \pi \pi \pi \pi }}_{\mathrm{\pi Έ\pi ½\pi Έ\pi \pi Έ\pi °\pi »}}\hfill \end{array}$

Proof 156 Similar to PropositionΒ 151.

Proposition 159

$\begin{array}{cc}& \mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\beta §\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }:\hfill \\ & {\mathrm{\pi \pi \pi }}_{2}\beta €\left(\stackrel{Β―}{\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}}>0\right)+β\frac{\mathrm{\Xi ±}}{\mathrm{\Xi ²}}β+\left(\mathrm{\Xi ±}\mathrm{mod}\mathrm{\Xi ²}\beta ₯max\left(1,\underset{Μ²}{\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}}\right)\right)\hfill \\ & \left\{\begin{array}{c}\beta ’\mathrm{\Xi ±}=max\left(0,{\mathrm{\pi \pi \pi \pi \pi \pi \pi }}_{\mathrm{\pi Έ\pi ½\pi Έ\pi \pi Έ\pi °\pi »}}-max\left(1,\underset{Μ²}{\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}}\right)-max\left(1,\underset{Μ²}{\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}}\right)\right),\hfill \\ \beta ’\mathrm{\Xi ²}=max\left(1,\underset{Μ²}{\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}}\right)+max\left(1,\underset{Μ²}{\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}}\right).\hfill \end{array}\right\\hfill \end{array}$

Proof 157 Similar to PropositionΒ 152.

Proposition 160

$\begin{array}{cc}& \mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\beta §\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }:\hfill \\ & \mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}Β·max\left(0,{\mathrm{\pi \pi \pi }}_{2}-1\right)+\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}+\hfill \\ & \mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}Β·{\mathrm{\pi \pi \pi }}_{2}+\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}\beta ₯{\mathrm{\pi \pi \pi \pi \pi \pi \pi }}_{\mathrm{\pi Έ\pi ½\pi Έ\pi \pi Έ\pi °\pi »}}\hfill \end{array}$

Proof 158 Similar to PropositionΒ 153.

Proposition 161

$\begin{array}{cc}& \mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\beta §\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }:\hfill \\ & {\mathrm{\pi \pi \pi }}_{2}\beta ₯\left(\stackrel{Β―}{\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}}<{\mathrm{\pi \pi \pi \pi \pi \pi \pi }}_{\mathrm{\pi Έ\pi ½\pi Έ\pi \pi Έ\pi °\pi »}}\right)+β\frac{\mathrm{\Xi ±}}{\mathrm{\Xi ²}}β+\left(\mathrm{\Xi ±}\mathrm{mod}\mathrm{\Xi ²}>\stackrel{Β―}{\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}}\right)\hfill \\ & \left\{\begin{array}{c}\beta ’\mathrm{\Xi ±}=max\left(0,{\mathrm{\pi \pi \pi \pi \pi \pi \pi }}_{\mathrm{\pi Έ\pi ½\pi Έ\pi \pi Έ\pi °\pi »}}-\stackrel{Β―}{\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}}-\stackrel{Β―}{\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}},\hfill \\ \beta ’\mathrm{\Xi ²}=max\left(1,\stackrel{Β―}{\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}}\right)+max\left(1,\stackrel{Β―}{\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}}\right).\hfill \end{array}\right\\hfill \end{array}$

Proof 159 Similar to PropositionΒ 154.

Proposition 162

Proof 160 Similar to PropositionΒ 155.

Proposition 163

Proof 161 Similar to PropositionΒ 156.

Proposition 164

Proof 162 Similar to PropositionΒ 157.