### 4.3.4.3. three parameters/one final graph

Proposition 68

Proof 68 (89) $n-1$ arcs are needed to connect $n$ $\left(n>1\right)$ vertices that all belong to a given connected component. Since we have two connected components, which respectively have $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }$ and $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }$ vertices, this leads to the previous inequality. When $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }$ is equal to one we need an extra arc.

Proposition 69

Proposition 70

Proposition 71

$\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }\beta €max\left(\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi },\mathrm{\pi \pi \pi \pi \pi \pi \pi }-max\left(1,\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }\right)\right)$

Proof 71 On the one hand, if $\mathrm{\pi \pi \pi }\beta €1$, we have that $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }\beta €\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }$. On the other hand, if $\mathrm{\pi \pi \pi }>1$, we have that $\mathrm{\pi \pi \pi \pi \pi \pi \pi }\beta ₯max\left(1,\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }\right)+\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }$ (i.e.,Β $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }\beta €\mathrm{\pi \pi \pi \pi \pi \pi \pi }-max\left(1,\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }\right)$). The result is obtained by taking the maximum value of the right hand side of the two inequalities.

Proposition 72

$\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }\beta \left[\mathrm{\pi \pi \pi \pi \pi \pi \pi }-max\left(1,\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }\right)+1,\mathrm{\pi \pi \pi \pi \pi \pi \pi }-1\right]$

Proof 72 On the one hand, if $\mathrm{\pi \pi \pi }\beta €1$, we have that $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }\beta ₯\mathrm{\pi \pi \pi \pi \pi \pi \pi }$. On the other hand, if $\mathrm{\pi \pi \pi }>1$, we have that $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }+max\left(1,\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }\right)\beta €\mathrm{\pi \pi \pi \pi \pi \pi \pi }$ (i.e.,Β $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }\beta €\mathrm{\pi \pi \pi \pi \pi \pi \pi }-max\left(1,\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }\right)$). The result follows.

Proposition 73

$\mathrm{\pi \pi \pi \pi \pi \pi \pi }\beta \left[\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }+1,\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }+\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }-1\right]$

Proof 73 On the one hand, if $\mathrm{\pi \pi \pi }\beta €1$, we have that $\mathrm{\pi \pi \pi \pi \pi \pi \pi }\beta €\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }$. On the other hand, if $\mathrm{\pi \pi \pi }>1$, we have that $\mathrm{\pi \pi \pi \pi \pi \pi \pi }\beta ₯\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }+\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }$. Since $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }\beta €\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }+\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }$ the result follows.

Proposition 74

$\begin{array}{cc}& \mathrm{\pi \pi }\stackrel{Β―}{\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }}>0\hfill \\ & \mathrm{\pi \pi \pi \pi }{k}_{\mathrm{\pi \pi \pi }}=β\frac{\underset{Μ²}{\mathrm{\pi \pi \pi \pi \pi \pi \pi }}+\stackrel{Β―}{\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }}}{\stackrel{Β―}{\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }}}β\mathrm{\pi \pi \pi \pi }{k}_{\mathrm{\pi \pi \pi }}=1\hfill \end{array}$

Proof 74 We make the proof for $k\beta \mathrm{\beta }$ (the interval is only used for restricting the number of intervals to check). We have that $\mathrm{\pi \pi \pi \pi \pi \pi \pi }\beta \left[kΒ·\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi },kΒ·\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }\right]$. A forbidden interval $\left[kΒ·\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }+1,\left(k+1\right)Β·\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }-1\right]$ corresponds to an interval between the end of interval $\left[kΒ·\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi },kΒ·\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }\right]$ and the start of the next interval $\left[\left(k+1\right)Β·\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi },\left(k+1\right)Β·\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }\right]$. Since all intervals $\left[iΒ·\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi },iΒ·\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }\right]$ $\left(i end before $kΒ·\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }$ and since all intervals $\left[jΒ·\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi },jΒ·\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }\right]$ $\left(j>k\right)$ start after $\left(k+1\right)Β·\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }$, they do not use any value in $\left[kΒ·\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }+1,\left(k+1\right)Β·\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }-1\right]$.

Proposition 75

$\mathrm{\pi \pi \pi \pi }\beta €\mathrm{\pi \pi \pi }Β·\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}^{2}$

Proof 75 On the one hand, (97) holds since the maximum number of arcs is achieved by taking $\mathrm{\pi \pi \pi }$ connected components where each connected component is a clique involving $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }$ vertices. On the other hand, (98) holds since a tree of $n$ vertices has $n-1$ arcs.

Proposition 76

$\mathrm{\pi \pi \pi \pi }\beta ₯\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }+\mathrm{\pi \pi \pi }-2$

Proof 76 The minimum number of arcs is achieved by taking one connected component with $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }$ vertices and $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }-1$ arcs as well as $\mathrm{\pi \pi \pi }-1$ connected components with one single vertex and a loop.

Proposition 77

$\mathrm{\pi \pi \pi \pi }\beta €\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}^{2}Β·β\frac{\mathrm{\pi \pi \pi \pi \pi \pi \pi }}{max\left(1,\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }\right)}β+{\left(\mathrm{\pi \pi \pi \pi \pi \pi \pi }\mathrm{mod}max\left(1,\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }\right)\right)}^{2}$

Proof 77 If $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }=0$ we get $\mathrm{\pi \pi \pi \pi }\beta €0$ which holds since the set of vertices is empty. We now assume that $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }>0$. We first begin with the following claim:

let $G$ be a graph such that $V\left(G\right)-\mathrm{\pi \pi \pi }\left(G,\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }\left(G\right)\right)*\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }\left(G\right)\beta ₯\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }\left(G\right)$, then there exists a graph ${G}^{\text{'}}$ such that $V\left({G}^{\text{'}}\right)=V\left(G\right)$, $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }\left({G}^{\text{'}}\right)=\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }\left(G\right)$, $\mathrm{\pi \pi \pi }\left({G}^{\text{'}},\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }\left({G}^{\text{'}}\right)\right)=\mathrm{\pi \pi \pi }\left(G,\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }\left(G\right)\right)+1$ and $|E\left(G\right)|\beta €|E\left({G}^{\text{'}}\right)|$.

Proof of the claim

Let ${\left({C}_{i}\right)}_{i\beta \left[n\right]}$ be the connected components of $G$ on less than $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }\left(G\right)$ vertices and such that $|{C}_{i}|\beta ₯|{C}_{i+1}|$. By hypothesis there exists $k\beta €n$ such that $|{\beta }_{i=1}^{k-1}{C}_{i}|<\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }\left(G\right)$ and $|{\beta }_{i=1}^{k}{C}_{i}|\beta ₯\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }\left(G\right)$.

• Either $|{\beta }_{i=1}^{k}{C}_{i}|=\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }\left(G\right)$, and then with ${G}^{\text{'}}$ such that ${G}^{\text{'}}$ restricted to the ${\beta }_{i=1}^{k}{C}_{i}$ be a complete graph and ${G}^{\text{'}}$ restricted to $V\left(G\right)-{\beta }_{i=1}^{k}{C}_{i}$ being exactly $G$ restricted to $V\left(G\right)-{\beta }_{i=1}^{k}{C}_{i}$ we obtain the claim.

• Or $|{\beta }_{i=1}^{k}{C}_{i}|>\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }\left(G\right)$. Then ${C}_{k}={C}_{k}^{1}\beta {C}_{k}^{2}$ such that $|\left({\beta }_{i=1}^{k-1}{C}_{i}\right)\beta ͺ{C}_{k}^{1}|=\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }\left(G\right)$ and $|{C}_{k}^{2}|<|{C}_{1}|$ (notice that $k\beta ₯2$). Then with ${G}^{\text{'}}$ such that ${G}^{\text{'}}$ restricted to $\left({\beta }_{i=1}^{k-1}{C}_{i}\right)\beta ͺ{C}_{k}^{1}$ is a complete graph and ${G}^{\text{'}}$ restricted to $V\left(G\right)-\left(\left({\beta }_{i=1}^{k-1}{C}_{i}\right)\beta ͺ{C}_{k}^{1}\right)$ is exactly $G$ restricted to $V\left(G\right)-\left(\left({\beta }_{i=1}^{k-1}{C}_{i}\right)\beta ͺ{C}_{k}^{1}\right)$ we obtain the claim.

End of proof of the claim

We prove by induction on $r\left(G\right)=β\frac{\mathrm{\pi \pi \pi \pi \pi \pi \pi }\left(G\right)}{\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }\left(G\right)}β-\mathrm{\pi \pi \pi }\left(G,\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }\left(G\right)\right)$, where $G$ is any graph. For $r\left(G\right)=0$ the result holds (see Prop 44). Otherwise, since $r\left(G\right)>0$ we have that $V\left(G\right)-\mathrm{\pi \pi \pi }\left(G,\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }\left(G\right)\right)*\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }\left(G\right)\beta ₯\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }\left(G\right)$, by the previous claim there exists ${G}^{\text{'}}$ with the same number of vertices and the same number of vertices in the largest connected component, such that $r\left({G}^{\text{'}}\right)=r\left(G\right)-1$. Consequently the result holds by induction.

Proposition 78

$\mathrm{\pi \pi \pi \pi }\beta ₯\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }-1+β\frac{\mathrm{\pi \pi \pi \pi \pi \pi \pi }-\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }+1}{2}β$

Proof 78 Let $G$ be a graph, let $X$ be a maximal size connected component of $G$, then we have $G=G\left[X\right]\beta G\left[V\left(G\right)-X\right]$. On the one hand, as $G\left[X\right]$ is connected, by setting $\mathrm{\pi \pi \pi }=1$ inΒ 143 of PropositionΒ 99, we have $|E\left(G\left[X\right]\right)\beta ₯|X|-1$, on the other hand, by PropositionΒ 52, $|E\left(G\left[V\left(G\right)-X\right]\right)|\beta ₯β\frac{|V\left(G\right)-X|}{2}β$. Thus the result follows.

Proposition 79

$\mathrm{\pi \pi \pi \pi \pi }\beta €\mathrm{\pi \pi \pi }Β·max\left(0,\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }-1\right)$

Proof 79 Since a connected component contains at most $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }$ vertices and since it does not contain any isolated vertex a connected component involves at most $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }-1$ sinks. Thus the result follows.

Proposition 80

$\mathrm{\pi \pi \pi \pi \pi \pi \pi }\beta €\mathrm{\pi \pi \pi }Β·max\left(0,\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }-1\right)$

Proof 80 Similar to PropositionΒ 79.

Proposition 81

$\mathrm{\pi \pi \pi \pi \pi \pi \pi }\beta €\mathrm{\pi \pi \pi }Β·\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }$

Proof 81 The number of vertices is less than or equal to the number of connected components multiplied by the largest number of vertices in a connected component.

Proposition 82

$\mathrm{\pi \pi \pi \pi \pi \pi \pi }\beta ₯\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }+max\left(0,\mathrm{\pi \pi \pi }-1\right)$

$\mathrm{\pi \pi }\mathrm{\pi \pi \pi \pi }:\mathrm{\pi \pi \pi \pi \pi \pi \pi }\beta ₯\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }+max\left(0,2Β·\mathrm{\pi \pi \pi }-2\right)$

Proof 82 (105) The minimum number of vertices according to a fixed number of connected components $\mathrm{\pi \pi \pi }$ such that one of the connected component contains $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }$ vertices is obtained as follows: we get $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }$ vertices from the connected component involving $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }$ vertices and one vertex for each remaining connected component.

Proposition 83

Proof 83 (107) In a strongly connected component at least one arc has to leave each arc. Since we have two strongly connected components, which respectively have $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }$ and $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }$ vertices, this leads to the previous inequality.

Proposition 84

Proposition 85

Proposition 86

$\begin{array}{cc}& \mathrm{\pi \pi }\stackrel{Β―}{\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }}>0\hfill \\ & \mathrm{\pi \pi \pi \pi }{k}_{\mathrm{\pi \pi \pi }}=β\frac{\underset{Μ²}{\mathrm{\pi \pi \pi \pi \pi \pi \pi }}+\stackrel{Β―}{\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }}}{\stackrel{Β―}{\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }}}β\mathrm{\pi \pi \pi \pi }{k}_{\mathrm{\pi \pi \pi }}=1\hfill \end{array}$

Proof 86 Similar to PropositionΒ 74.

Proposition 87

$\mathrm{\pi \pi \pi \pi \pi \pi \pi }\beta €\mathrm{\pi \pi \pi }Β·\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }$

Proof 87 The largest number of vertices is obtained by putting within each connected component the number of vertices of the largest strongly connected component.

Proposition 88

$\mathrm{\pi \pi \pi \pi \pi \pi \pi }\beta €\mathrm{\pi \pi \pi \pi }Β·\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }$

Proof 88 Since each strongly connected component contains at most $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }$ vertices the total number of vertices is less than or equal to $\mathrm{\pi \pi \pi \pi }Β·\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }$.

Proposition 89

$\mathrm{\pi \pi \pi \pi \pi \pi \pi }\beta ₯\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }+max\left(0,\mathrm{\pi \pi \pi \pi }-1\right)$

$\mathrm{\pi \pi }\mathrm{\pi \pi \pi \pi }:\mathrm{\pi \pi \pi \pi \pi \pi \pi }\beta ₯\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }+max\left(0,2Β·\mathrm{\pi \pi \pi \pi }-2\right)$

Proof 89 (114) The minimum number of vertices according to a fixed number of strongly connected components $\mathrm{\pi \pi \pi \pi }$ such that one of them contains $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }$ vertices is equal to $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }+max\left(0,\mathrm{\pi \pi \pi \pi }-1\right)$.

Proposition 90

$\mathrm{\pi \pi \pi \pi }\beta €\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}^{2}+{\left(\mathrm{\pi \pi \pi \pi \pi \pi \pi }-\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }\right)}^{2}$

Proof 90 (116) The maximum number of vertices according to a fixed number of vertices $\mathrm{\pi \pi \pi \pi \pi \pi \pi }$ and to the fact that there is a connected component with $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }$ vertices is obtained by:

Proposition 91

$\begin{array}{cc}& \mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }>1\beta \hfill \\ & \mathrm{\pi \pi \pi \pi }\beta ₯β\frac{\mathrm{\pi \pi \pi \pi \pi \pi \pi }}{\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }}βΒ·\left(\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }-1\right)+\mathrm{\pi \pi \pi \pi \pi \pi \pi }\mathrm{mod}\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }\hfill \end{array}$

Proof 91 Achieving the minimum number of arcs with a fixed number of vertices and with a minimum number of vertices greater than or equal to one in each connected component is achieved in the following way:

• Since the minimum number of arcs of a connected component of $n$ vertices is $n-1$, splitting a connected component into $k$ parts that all have more than one vertex saves $k-1$ arcs. Therefore we build a maximum number of connected components. Since each connected component has at least $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }$ vertices we get $β\frac{\mathrm{\pi \pi \pi \pi \pi \pi \pi }}{\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }}β$ connected components.

• Since we cannot build a connected component with the rest of the vertices (i.e.,Β $\mathrm{\pi \pi \pi \pi \pi \pi \pi }\mathrm{mod}\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }$ vertices left) we have to incorporate them in the previous connected components and this costs one arc for each vertex.

When $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }=1$, note that PropositionΒ 52 provides a lower bound on the number of arcs.

Proposition 92

$\mathrm{\pi \pi \pi \pi \pi \pi \pi }\beta ₯\mathrm{\pi \pi \pi }Β·\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }$

Proof 92 The smallest number of vertices is obtained by taking all connected components to their minimum number of vertices $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }$.

Proposition 93

$\mathrm{\pi \pi \pi \pi \pi \pi \pi }>\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }\beta \mathrm{\pi \pi \pi }\beta ₯2$

Proof 93 If all vertices do not fit within the smallest connected component then we have at least two connected components.

Proposition 94

$\mathrm{\pi \pi \pi \pi }\beta €{\mathrm{\pi \pi \pi \pi \pi \pi \pi }}^{2}+\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi \pi }}^{2}-\mathrm{\pi \pi \pi \pi \pi \pi \pi }Β·\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }$

Proof 94 Achieving the maximum number of arcs, provided that we have at least one strongly connected component with $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }$ vertices, is done by:

• Building a first strongly connected component ${\mathrm{\pi }}_{1}$ with $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }$ vertices and adding an arc between each pair of vertices of ${\mathrm{\pi }}_{1}$.

• Building a second strongly connected component ${\mathrm{\pi }}_{2}$ with $\mathrm{\pi \pi \pi \pi \pi \pi \pi }-\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }$ vertices and adding an arc between each pair of vertices of ${\mathrm{\pi }}_{2}$.

Finally, we add an arc from every vertex of ${\mathrm{\pi }}_{1}$ to every vertex of ${\mathrm{\pi }}_{2}$. This leads to a total number of arcs of $\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi \pi }}^{2}+{\left(\mathrm{\pi \pi \pi \pi \pi \pi \pi }-\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }\right)}^{2}+\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }Β·\left(\mathrm{\pi \pi \pi \pi \pi \pi \pi }-\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }\right)$.

Proposition 95

$\mathrm{\pi \pi \pi \pi \pi \pi \pi }\beta ₯\mathrm{\pi \pi \pi }Β·\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }$

Proof 95 The smallest number of vertices is obtained by putting within each connected component the number of vertices of the smallest strongly connected component.

Proposition 96

$\mathrm{\pi \pi \pi \pi \pi \pi \pi }\beta ₯\mathrm{\pi \pi \pi \pi }Β·\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }$

Proof 96 Since each strongly connected component contains at least $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }$ vertices the total number of vertices is greater than or equal to $\mathrm{\pi \pi \pi \pi }Β·\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }$.

Proposition 97

$\mathrm{\pi \pi \pi \pi \pi \pi \pi }>\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }\beta \mathrm{\pi \pi \pi \pi }\beta ₯2$

Proof 97 If all vertices do not fit within the smallest strongly connected component then we have at least two strongly connected components.

Proposition 98

$\mathrm{\pi \pi \pi \pi }\beta €{\left(\mathrm{\pi \pi \pi \pi \pi \pi \pi }-\mathrm{\pi \pi \pi }+1\right)}^{2}+\mathrm{\pi \pi \pi }-1$

Proof 98 (133) We proceed by induction on $T\left(G\right)=\mathrm{\pi \pi \pi \pi \pi \pi \pi }\left(G\right)-|X|-\left(\mathrm{\pi \pi \pi }\left(G\right)-1\right)$, where $X$ is any connected component of $G$ of maximum cardinality. For $T\left(G\right)=0$ then either $\mathrm{\pi \pi \pi }\left(G\right)=1$ and thus the formula is clearly true, or all the connected components of $G$, but possibly $X$, are reduced to one element. Since isolated vertices are not allowed, the formula holds.

Assume that $T\left(G\right)\beta ₯1$. Then there exists $Y$, a connected component of $G$ distinct from $X$, with more than one vertex. Let $y\beta Y$ and let ${G}^{\text{'}}$ be the graph such that $V\left({G}^{\text{'}}\right)=V\left(G\right)$ and $E\left({G}^{\text{'}}\right)$ is defined by:

• For all $Z$ connected components of $G$ distinct from $X$ and $Y$ we have ${G}^{\text{'}}\left[Z\right]=G\left[Z\right]$.

• With ${X}^{\text{'}}=X\beta ͺ\left\{y\right\}$ and ${Y}^{\text{'}}=Y-\left\{y\right\}$, we have ${G}^{\text{'}}\left[{Y}^{\text{'}}\right]=G\left[{Y}^{\text{'}}\right]$ and $E\left({G}^{\text{'}}\left[{X}^{\text{'}}\right]\right)=E\left(G\left[X\right]\right)\beta ͺ\left({\beta }_{x\beta {X}^{\text{'}}}\left\{\left(x,y\right),\left(y,x\right)\right\}\right)$.

Clearly $|E\left({G}^{\text{'}}\right)|-|E\left(G\right)|\beta ₯2Β·|X|+1-\left(2Β·|Y|-1\right)$ and since $X$ is of maximal cardinality the difference is strictly positive. Now as $\mathrm{\pi \pi \pi \pi \pi \pi \pi }\left({G}^{\text{'}}\right)=\mathrm{\pi \pi \pi \pi \pi \pi \pi }\left(G\right)$, $\mathrm{\pi \pi \pi }\left({G}^{\text{'}}\right)=\mathrm{\pi \pi \pi }\left(G\right)$ and as $T\left({G}^{\text{'}}\right)=T\left(G\right)-1$ the result holds by induction hypothesis.

Proposition 99

$\mathrm{\pi \pi \pi \pi }\beta ₯\mathrm{\pi \pi \pi \pi \pi \pi \pi }-\mathrm{\pi \pi \pi }$

$\begin{array}{cc}& \mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }:\mathrm{\pi \pi \pi }>0\beta \\ & \mathrm{\pi \pi \pi \pi }\beta ₯\left(\mathrm{\pi \pi \pi \pi \pi \pi \pi }\mathrm{mod}\mathrm{\pi \pi \pi }\right)Β·{\left(β\frac{\mathrm{\pi \pi \pi \pi \pi \pi \pi }}{\mathrm{\pi \pi \pi }}β+1\right)}^{2}+\\ & \left(\mathrm{\pi \pi \pi }-\mathrm{\pi \pi \pi \pi \pi \pi \pi }\mathrm{mod}\mathrm{\pi \pi \pi }\right)Β·{β\frac{\mathrm{\pi \pi \pi \pi \pi \pi \pi }}{\mathrm{\pi \pi \pi }}β}^{2}\end{array}$

Proof 99 (143) By induction of the number of vertices. The formula holds for one vertex. Let $G$ a graph with $n+1$ vertices $\left(n\beta ₯1\right)$. First assume there exists $x$ in $G$ such that $G-x$ has the same number of connected components than $G$. Since $\mathrm{\pi \pi \pi \pi }\left(G\right)\beta ₯\mathrm{\pi \pi \pi \pi }\left(G-x\right)+1$, and by induction hypothesis $\mathrm{\pi \pi \pi \pi }\left(G-x\right)\beta ₯\mathrm{\pi \pi \pi \pi \pi \pi \pi }\left(G-x\right)-\mathrm{\pi \pi \pi }\left(G-x\right)$ the result holds. Otherwise all connected components of $G$ are reduced to one vertex and the formula holds.

Proposition 100

$\mathrm{\pi \pi \pi \pi }\beta €\left(\mathrm{\pi \pi \pi \pi \pi \pi \pi }-\mathrm{\pi \pi \pi \pi }+1\right)Β·\mathrm{\pi \pi \pi \pi \pi \pi \pi }+\frac{\mathrm{\pi \pi \pi \pi }Β·\left(\mathrm{\pi \pi \pi \pi }-1\right)}{2}$

$\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }:\mathrm{\pi \pi \pi \pi }\beta €\mathrm{\pi \pi \pi \pi }-1+{\left(\mathrm{\pi \pi \pi \pi \pi \pi \pi }-\mathrm{\pi \pi \pi \pi }+1\right)}^{2}$

Proof 100 For provingΒ 145, it is easier to rewrite the formula as $\mathrm{\pi \pi \pi \pi }\beta €{\left(\mathrm{\pi \pi \pi \pi \pi \pi \pi }-\left(\mathrm{\pi \pi \pi \pi }-1\right)\right)}^{2}+\left(\mathrm{\pi \pi \pi }-1\right)Β·\left(\mathrm{\pi \pi \pi \pi \pi \pi \pi }-\left(\mathrm{\pi \pi \pi \pi }-1\right)\right)+\frac{\mathrm{\pi \pi \pi \pi }Β·\left(\mathrm{\pi \pi \pi \pi }-1\right)}{2}$. We proceed by induction on $T\left(G\right)=\mathrm{\pi \pi \pi \pi \pi \pi \pi }\left(G\right)-|X|-\left(\mathrm{\pi \pi \pi \pi }\left(G\right)-1\right)$, where $X$ is any strongly connected component of $G$ of maximum cardinality.

For $T\left(G\right)=0$ then either $\mathrm{\pi \pi \pi \pi }\left(G\right)=1$ and thus the formula is clearly true, or all the strongly connected components of $G$, but possibly $X$, are reduced to one element. Since the maximum number of arcs in a directed acyclic graph of $n$ vertices is $\frac{nΒ·\left(n+1\right)}{2}$, and as the subgraph of $G$ induced by all the strongly connected components of $G$ excepted $X$ is acyclic, the formula clearly holds.

Assume that $T\left(G\right)\beta ₯1$, let ${\left({X}_{i}\right)}_{i\beta I}$ be the family of strongly connected components of $G$, and let ${G}_{r}$ be the reduced graph of $G$ induced by ${\left({X}_{i}\right)}_{i\beta I}$ (that is $V\left({G}_{r}\right)=I$ and $\beta {i}_{1},{i}_{2}\beta I$, $\left({i}_{1},{i}_{2}\right)\beta E\left({G}_{r}\right)$ if and only if $\beta {x}_{1}\beta {X}_{{i}_{1}}$, $\beta {x}_{2}\beta {X}_{{i}_{2}}$ such that $\left({x}_{1},{x}_{2}\right)\beta E$). Consider ${G}^{\text{'}}$ such that $V\left({G}^{\text{'}}\right)=V\left(G\right)$ and $E\left({G}^{\text{'}}\right)$ is defined by:

• For all strongly connected components $Z$ of $G$ we have ${G}^{\text{'}}\left[Z\right]=G\left[Z\right]$.

• For $\mathrm{Ο}$ be any topological sort of ${G}_{r}$, $\beta {x}_{i}\beta {X}_{i}$, $\beta {x}_{j}\beta {X}_{j}$, $\left({x}_{i},{x}_{j}\right)\beta E\left({G}^{\text{'}}\right)$ whenever $i$ is less than $j$ with respect to $\mathrm{Ο}$.

Notice that ${G}^{\text{'}}$ satisfies the following properties: $T\left({G}^{\text{'}}\right)=T\left(G\right)$, $V\left({G}^{\text{'}}\right)=V\left(G\right)$, $\mathrm{\pi \pi \pi \pi }\left({G}^{\text{'}}\right)=\mathrm{\pi \pi \pi \pi }\left(G\right)$, $E\left(G\right)\beta E\left({G}^{\text{'}}\right)$, ${\left({X}_{i}\right)}_{i\beta I}$ is still the family of strongly connected components of ${G}^{\text{'}}$, and moreover, for every $i\beta I$ and every ${x}_{i}\beta {X}_{i}$ we have that ${x}_{i}$ is connected to any vertex outside ${X}_{i}$, that is the number of arcs incident to ${x}_{i}$ and incident to vertices outside ${X}_{i}$ is exactly $|V\left({G}^{\text{'}}\right)|-|{X}_{i}|$.

Now, as $T\left({G}^{\text{'}}\right)\beta ₯1$, there exists $Y$, a strongly connected component of ${G}^{\text{'}}$ distinct from $X$, with more than one vertex. Let $y\beta Y$ and let ${G}^{\text{'}\text{'}}$ be the graph such that $V\left({G}^{\text{'}\text{'}}\right)=V\left({G}^{\text{'}}\right)$ and $E\left({G}^{\text{'}\text{'}}\right)$ is defined by:

• ${G}^{\text{'}\text{'}}\left[V\left(G\right)-\left\{y\right\}\right]={G}^{\text{'}}\left[V\left(G\right)-\left\{y\right\}\right]$.

• With ${X}^{\text{'}}=X\beta ͺ\left\{y\right\}$, we have ${G}^{\text{'}\text{'}}\left[{Y}^{\text{'}}\right]={G}^{\text{'}}\left[{Y}^{\text{'}}\right]$ and $E\left({G}^{\text{'}\text{'}}\left[{X}^{\text{'}}\right]\right)=E\left({G}^{\text{'}}\left[X\right]\right)\beta ͺ\left({\beta }_{x\beta {X}^{\text{'}}}\left\{\left(x,y\right),\left(y,x\right)\right\}\right)$.

• Assume that $X={X}_{j}$ for $j\beta I$. Then $\beta i\beta I-\left\{j\right\}$, $\beta {x}_{i}\beta {X}_{i}$, $\left({x}_{i},y\right)\beta E\left({G}^{\text{'}\text{'}}\right)$ whenever $i$ is less than $j$ with respect to $\mathrm{Ο}$ and $\left(y,{x}_{i}\right)\beta E\left({G}^{\text{'}\text{'}}\right)$ whenever $j$ is less than $i$ with respect to $\mathrm{Ο}$.

Clearly $|E\left({G}^{\text{'}\text{'}}\right)|-|E\left({G}^{\text{'}}\right)|\beta ₯2|X|+1+|V\left({G}^{\text{'}}\right)|-|X|-\left(2Β·|Y|-1+|V\left({G}^{\text{'}}\right)|-|Y|\right)=|X|-|Y|+2$ and since $X$ is of maximal cardinality the difference is strictly positive. As $E\left(G\right)\beta E\left({G}^{\text{'}}\right)$, $|E\left({G}^{\text{'}\text{'}}\right)|-|E\left(G\right)|$ is also strictly positive. Now as $\mathrm{\pi \pi \pi \pi \pi \pi \pi }\left({G}^{\text{'}\text{'}}\right)=\mathrm{\pi \pi \pi \pi \pi \pi \pi }\left({G}^{\text{'}}\right)=\mathrm{\pi \pi \pi \pi \pi \pi \pi }\left(G\right)$, $\mathrm{\pi \pi \pi \pi }\left({G}^{\text{'}\text{'}}\right)=\mathrm{\pi \pi \pi \pi }\left({G}^{\text{'}}\right)=\mathrm{\pi \pi \pi \pi }\left(G\right)$ and as $T\left({G}^{\text{'}\text{'}}\right)=T\left({G}^{\text{'}}\right)-1=T\left(G\right)-1$ the result holds by induction hypothesis.

Proposition 101

$\mathrm{\pi \pi \pi \pi }\beta ₯\mathrm{\pi \pi \pi \pi \pi \pi \pi }-β\frac{\mathrm{\pi \pi \pi \pi }-1}{2}β$

$\begin{array}{cc}& \mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }:\mathrm{\pi \pi \pi \pi }>0\beta \\ & \mathrm{\pi \pi \pi \pi }\beta ₯\left(\mathrm{\pi \pi \pi \pi \pi \pi \pi }\mathrm{mod}\mathrm{\pi \pi \pi \pi }\right)Β·{\left(β\frac{\mathrm{\pi \pi \pi \pi \pi \pi \pi }}{\mathrm{\pi \pi \pi \pi }}β+1\right)}^{2}+\\ & \left(\mathrm{\pi \pi \pi \pi }-\mathrm{\pi \pi \pi \pi \pi \pi \pi }\mathrm{mod}\mathrm{\pi \pi \pi \pi }\right)Β·{β\frac{\mathrm{\pi \pi \pi \pi \pi \pi \pi }}{\mathrm{\pi \pi \pi \pi }}β}^{2}\end{array}$

Proof 101 For proving partΒ 147 of PropositionΒ 101 we proceed by induction on $\mathrm{\pi \pi \pi \pi }\left(G\right)$. If $\mathrm{\pi \pi \pi \pi }\left(G\right)=1$ then, we have $\mathrm{\pi \pi \pi \pi }\left(G\right)\beta ₯\mathrm{\pi \pi \pi \pi \pi \pi \pi }\left(G\right)$ (i.e.,Β for one vertex this is true since every vertex has at least one arc, otherwise every vertex $v$ has an arc arriving on $v$ as well as an arc starting from $v$, thus we have $\mathrm{\pi \pi \pi \pi }\beta ₯\frac{2Β·\mathrm{\pi \pi \pi \pi \pi \pi \pi }}{2}$). If $\mathrm{\pi \pi \pi \pi }\left(G\right)>1$ let $X$ be a strongly connected component of $G$. Then $\mathrm{\pi \pi \pi \pi }\left(G\right)\beta ₯\mathrm{\pi \pi \pi \pi }\left(G\left[V\left(G\right)-X\right]\right)+\mathrm{\pi \pi \pi \pi }\left(G\left[X\right]\right)$. By induction hypothesis $\mathrm{\pi \pi \pi \pi }\left(G\left[V\left(G\right)-X\right]\right)\beta ₯|V\left(G\right)-X|-β\frac{\mathrm{\pi \pi \pi \pi }\left(G\left[V\left(G\right)-X\right]\right)-1}{2}β$, thus $\mathrm{\pi \pi \pi \pi }\left(G\left[V\left(G\right)-X\right]\right)\beta ₯|V\left(G\right)-X|-β\frac{\left(\mathrm{\pi \pi \pi \pi }\left(G\right)-1\right)-1}{2}β$. Since $\mathrm{\pi \pi \pi \pi }\left(G\left[X\right]\right)\beta ₯|X|$ we obtain $\mathrm{\pi \pi \pi \pi }\left(G\right)\beta ₯|V\left(G\right)|-β\frac{\left(\mathrm{\pi \pi \pi \pi }\left(G\right)-1\right)-1}{2}β$, and thus the result holds.

Proposition 102

$\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }:\mathrm{\pi \pi \pi \pi \pi \pi \pi }>0\beta \mathrm{\pi \pi \pi \pi }\beta ₯β\frac{{\mathrm{\pi \pi \pi \pi \pi \pi \pi }}^{2}}{\mathrm{\pi \pi \pi \pi }}β$

Proof 102 As shown inΒ [BessiereHebrardHnichKiziltanWalsh05], a lower bound for the minimum number of equivalence classes (e.g.,Β strongly connected components) is the independence number of the graph and the right -hand side of PropositionΒ 102 corresponds to a lower bound of the independence number proposed by TurΓ‘n

Proposition 103

$\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }:\mathrm{\pi \pi \pi \pi \pi \pi \pi }>0\beta \mathrm{\pi \pi \pi \pi }\beta ₯β\frac{2Β·\mathrm{\pi \pi \pi \pi \pi \pi \pi }-\frac{\mathrm{\pi \pi \pi \pi }-\mathrm{\pi \pi \pi \pi \pi \pi \pi }}{β\frac{\mathrm{\pi \pi \pi \pi }-\mathrm{\pi \pi \pi \pi \pi \pi \pi }}{\mathrm{\pi \pi \pi \pi \pi \pi \pi }}β}}{β\frac{\mathrm{\pi \pi \pi \pi }-\mathrm{\pi \pi \pi \pi \pi \pi \pi }}{\mathrm{\pi \pi \pi \pi \pi \pi \pi }}β+1}β$

Proof 103 See

Proposition 104

$\mathrm{\pi \pi \pi \pi }\beta €\left(\mathrm{\pi \pi \pi \pi \pi \pi \pi }-\mathrm{\pi \pi \pi \pi \pi }\right)Β·\mathrm{\pi \pi \pi \pi \pi \pi \pi }$

Proof 104 The maximum number of arcs is achieved by the following pattern: for all non -sink vertices we have an arc to all vertices.

Proposition 105

$\mathrm{\pi \pi \pi \pi }\beta ₯\mathrm{\pi \pi \pi \pi \pi }+max\left(0,\mathrm{\pi \pi \pi \pi \pi \pi \pi }-2Β·\mathrm{\pi \pi \pi \pi \pi }\right)$

Proof 105 Recall that for $x\beta V\left(G\right)$, we have that ${d}_{G}^{+}\left(x\right)+{d}_{G}^{-}\left(x\right)\beta ₯1$. If $x$ is a sink then ${d}_{G}^{-}\left(x\right)\beta ₯1$, consequently $\mathrm{\pi \pi \pi \pi }\left(G\right)\beta ₯\mathrm{\pi \pi \pi \pi \pi }\left(G\right)$. If $x$ is not a sink then ${d}_{G}^{+}\left(x\right)\beta ₯1$, consequently $\mathrm{\pi \pi \pi \pi }\left(G\right)\beta ₯|V\left(G\right)|-\mathrm{\pi \pi \pi \pi \pi }\left(G\right)$.

Proposition 106

$\mathrm{\pi \pi \pi \pi }\beta €\left(\mathrm{\pi \pi \pi \pi \pi \pi \pi }-\mathrm{\pi \pi \pi \pi \pi \pi \pi }\right)Β·\mathrm{\pi \pi \pi \pi \pi \pi \pi }$

Proof 106 The maximum number of arcs is achieved by the following pattern: for all non -source vertices we have an arc from all vertices.

Proposition 107

$\mathrm{\pi \pi \pi \pi }\beta ₯\mathrm{\pi \pi \pi \pi \pi \pi \pi }+max\left(0,\mathrm{\pi \pi \pi \pi \pi \pi \pi }-2Β·\mathrm{\pi \pi \pi \pi \pi \pi \pi }\right)$

Proof 107 Similar to PropositionΒ 105.

Proposition 108

$\mathrm{\pi \pi \pi \pi }\beta ₯\mathrm{\pi \pi \pi \pi \pi }+\mathrm{\pi \pi \pi \pi \pi \pi \pi }$

Proof 108 Since sinks and sources cannot belong to a circuit and since they cannot coincide (i.e.,Β because isolated vertices are not allowed) the result follows.

Proposition 109

$\mathrm{\pi \pi \pi \pi \pi \pi \pi }\beta ₯\mathrm{\pi \pi \pi \pi \pi }+\mathrm{\pi \pi \pi \pi \pi \pi \pi }$

Proof 109 No vertex can be both a source and a sink (isolated vertices are removed).