### 4.3.4.6. two parameters/two final graphs

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

Proposition 125

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

Proof 123 We show that the conjunction $\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}\beta ₯{\mathrm{\pi \pi \pi \pi \pi \pi \pi }}_{\mathrm{\pi Έ\pi ½\pi Έ\pi \pi Έ\pi °\pi »}}-\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}$ and $\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}\beta €\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}-1$ leads to a contradiction.

Since $\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}\beta €\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}-1$ we have that and the minimum required size for the different groups is $\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}+1+\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}$. This minimum required size should not exceed the number of vertices ${\mathrm{\pi \pi \pi \pi \pi \pi \pi }}_{\mathrm{\pi Έ\pi ½\pi Έ\pi \pi Έ\pi °\pi »}}$ of the initial graph. But since, by hypothesis, $\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}\beta ₯{\mathrm{\pi \pi \pi \pi \pi \pi \pi }}_{\mathrm{\pi Έ\pi ½\pi Έ\pi \pi Έ\pi °\pi »}}-\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}$, this is impossible.

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

Proposition 126

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

Proof 124 Similar to PropositionΒ 125.

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

Proposition 127

$\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }:\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}<{\mathrm{\pi \pi \pi \pi \pi \pi \pi }}_{\mathrm{\pi Έ\pi ½\pi Έ\pi \pi Έ\pi °\pi »}}\beta {\mathrm{\pi \pi \pi }}_{2}>0$

$\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }:\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}<{\mathrm{\pi \pi \pi \pi \pi \pi \pi }}_{\mathrm{\pi Έ\pi ½\pi Έ\pi \pi Έ\pi °\pi »}}\beta {\mathrm{\pi \pi \pi }}_{2}>0$

Proof 125 (186) Since we have the precondition $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }$, we know that each vertex of the initial graph belongs to the first or to the second final graphs (but not to both).

1. On the one hand, if the largest connected component of the first final graph cannot contain all the vertices of the initial graph, then the second final graph has at least one connected component.

2. On the other hand, if the second final graph has at least one connected component then the largest connected component of the first final graph cannot be equal to the initial graph.

(187) holds for a similar reason.

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

Proposition 128

$\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }:\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}<{\mathrm{\pi \pi \pi \pi \pi \pi \pi }}_{\mathrm{\pi Έ\pi ½\pi Έ\pi \pi Έ\pi °\pi »}}\beta {\mathrm{\pi \pi \pi }}_{1}>0$

$\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }:\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}<{\mathrm{\pi \pi \pi \pi \pi \pi \pi }}_{\mathrm{\pi Έ\pi ½\pi Έ\pi \pi Έ\pi °\pi »}}\beta {\mathrm{\pi \pi \pi }}_{1}>0$

Proof 126 Similar to PropositionΒ 127.

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

Proposition 129

$\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }:\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{1}<{\mathrm{\pi \pi \pi \pi \pi \pi \pi }}_{\mathrm{\pi Έ\pi ½\pi Έ\pi \pi Έ\pi °\pi »}}\beta {\mathrm{\pi \pi \pi }}_{2}>0$

Proof 127 Since we have the precondition $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }$, we know that each vertex of the initial graph belongs to the first or to the second final graphs (but not to both).

1. On the one hand, if the smallest connected component of the first final graph cannot contain all the vertices of the initial graph, then the second final graph has at least one connected component.

2. On the other hand, if the second final graph has at least one connected component then the smallest connected component of the first final graph cannot be equal to the initial graph.

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

Proposition 130

$\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }:\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}_{2}<{\mathrm{\pi \pi \pi \pi \pi \pi \pi }}_{\mathrm{\pi Έ\pi ½\pi Έ\pi \pi Έ\pi °\pi »}}\beta {\mathrm{\pi \pi \pi }}_{1}>0$

Proof 128 Similar to PropositionΒ 129.

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

Proposition 131

Proof 129 Holds since each arc of the initial graph belongs to one of the two final graphs and since the initial graph has ${\mathrm{\pi \pi \pi \pi \pi \pi \pi }}_{\mathrm{\pi Έ\pi ½\pi Έ\pi \pi Έ\pi °\pi »}}-1$ arcs.

${\mathrm{\pi \pi \pi }}_{1}$, ${\mathrm{\pi \pi \pi }}_{2}$

Proposition 132

$\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\beta §\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }:|{\mathrm{\pi \pi \pi }}_{1}-{\mathrm{\pi \pi \pi }}_{2}|\beta €1$

Proof 130 Holds because the two initial graphs correspond to a path and because consecutive connected components do not come from the same graph constraint.

Proposition 133

Proof 131 Holds because the initial graph is a path.

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

Proposition 134

$\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }:{\mathrm{\pi \pi \pi \pi \pi \pi \pi }}_{1}+{\mathrm{\pi \pi \pi \pi \pi \pi \pi }}_{2}={\mathrm{\pi \pi \pi \pi \pi \pi \pi }}_{\mathrm{\pi Έ\pi ½\pi Έ\pi \pi Έ\pi °\pi »}}$

Proof 132 By definition of $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }$ each vertex of the initial graph belongs to one of the two final graphs (but not to both).