### 4.3.4.2. two parameters/one final graph

Proposition 14

$\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }=0\beta \mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }=0$

Proposition 15

$\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }\beta €\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }$

Proof 15 $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }$ is a lower bound of the size of the largest connected component since the largest strongly connected component is for sure included within a connected component.

Proposition 16

$\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }=0\beta \mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }=0$

Proposition 17

$\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }\beta €\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }$

Proposition 18

$\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }=0\beta \mathrm{\pi \pi \pi \pi }=0$

Proposition 19

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

$\mathrm{\pi \pi ’\pi \pi \pi \pi \pi \pi \pi }:\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }>0\beta \mathrm{\pi \pi \pi \pi }\beta ₯max\left(1,2Β·\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }-2\right)$

$\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }:\mathrm{\pi \pi \pi \pi }\beta ₯\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}^{2}$

Proof 19 Β

(19) $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }-1$ arcs are needed to connect $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }$ vertices that belong to a given connected component containing at least two vertices. And one arc is required for a connected component containing a single vertex.

(20) Similarly, when the graph is symmetric, $2Β·\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }-2$ arcs are needed to connect $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }$ vertices that belong to a given connected component containing at least two vertices.

(21) Finally, when the graph is reflexive, symmetric and transitive, $\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}^{2}$ arcs are needed to connect $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }$ vertices that belong to a given connected component.

(22) When the initial graph corresponds to a path, the minimum number of arcs of a connected component involving $n$ vertices is equal to $n-1$.

Proposition 20

$\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }=0\beta \mathrm{\pi \pi \pi \pi \pi }=0$

Proposition 21

$\mathrm{\pi \pi \pi \pi \pi }\beta ₯1\beta \mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }\beta ₯2$

Proof 21 Since we do not have any isolated vertex a sink is connected to at least one other vertex. Therefore, if the graph has a sink, there exists at least one connected component with at least two vertices.

Proposition 22

$\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }=0\beta \mathrm{\pi \pi \pi \pi \pi \pi \pi }=0$

Proposition 23

$\mathrm{\pi \pi \pi \pi \pi \pi \pi }\beta ₯1\beta \mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }\beta ₯2$

Proof 23 Since we do not have any isolated vertex a source is connected to at least one other vertex. Therefore, if the graph has a source, there exists at least one connected component with at least two vertices.

Proposition 24

$\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }=0\beta \mathrm{\pi \pi \pi \pi \pi \pi \pi }=0$

Proposition 25

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

Proof 25

Proposition 26

$\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }=0\beta \mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }=0$

Proposition 27

$\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }\beta €\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }$

Proposition 28

$\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }=0\beta \mathrm{\pi \pi \pi \pi }=0$

Proposition 29

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

$\mathrm{\pi \pi ’\pi \pi \pi \pi \pi \pi \pi }:\mathrm{\pi \pi \pi \pi }\beta ₯2Β·\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }$

$\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }:\mathrm{\pi \pi \pi \pi }\beta ₯\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi \pi }}^{2}$

Proof 29 (32) In a strongly connected component at least one arc has to leave each vertex. Since we have at least one strongly connected component of $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }$ vertices this leads to the previous inequality.

Proposition 30

$\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }=0\beta \mathrm{\pi \pi \pi \pi \pi \pi \pi }=0$

Proposition 31

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

Proof 31 By definition of $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }$.

Proposition 32

$\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }=0\beta \mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }=0$

Proposition 33

$\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }\beta ₯\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }$

Proof 33 By construction $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }$ is an upper bound of the number of vertices of the smallest strongly connected component.

Proposition 34

$\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }=0\beta \mathrm{\pi \pi \pi \pi }=0$

Proposition 35

$\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }>0\beta \mathrm{\pi \pi \pi \pi }\beta ₯max\left(1,\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }-1\right)$

$\mathrm{\pi \pi ’\pi \pi \pi \pi \pi \pi \pi }:\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }>0\beta \mathrm{\pi \pi \pi \pi }\beta ₯max\left(1,2Β·\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }-2\right)$

$\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }:\mathrm{\pi \pi \pi \pi }\beta ₯\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi }}^{2}$

Proof 35 Similar to PropositionΒ 19.

Proposition 36

$\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }:\left(\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }+1\right)Β·\mathrm{\pi \pi \pi }\beta €{\mathrm{\pi \pi \pi \pi \pi \pi \pi }}_{\mathrm{\pi Έ\pi ½\pi Έ\pi \pi Έ\pi °\pi »}}+1$

Proof 36 By definition of $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }$.

Proposition 37

$\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }=0\beta \mathrm{\pi \pi \pi \pi \pi \pi \pi }=0$

Proposition 38

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

Proof 38

Proposition 39

$\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }\beta \left[min\left(β\frac{\mathrm{\pi \pi \pi \pi \pi \pi \pi }}{2}β,β\frac{{\mathrm{\pi \pi \pi \pi \pi \pi \pi }}_{\mathrm{\pi Έ\pi ½\pi Έ\pi \pi Έ\pi °\pi »}}-1}{2}β\right)+1,\mathrm{\pi \pi \pi \pi \pi \pi \pi }-1\right]$

Proof 39 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 $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }+\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }\beta €\mathrm{\pi \pi \pi \pi \pi \pi \pi }$ and that $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }+\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }+1\beta €{\mathrm{\pi \pi \pi \pi \pi \pi \pi }}_{\mathrm{\pi Έ\pi ½\pi Έ\pi \pi Έ\pi °\pi »}}$, which by isolating $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }$ and by grouping the two inequalities leads to $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }\beta €min\left(β\frac{\mathrm{\pi \pi \pi \pi \pi \pi \pi }}{2}β,β\frac{{\mathrm{\pi \pi \pi \pi \pi \pi \pi }}_{\mathrm{\pi Έ\pi ½\pi Έ\pi \pi Έ\pi °\pi »}}-1}{2}β\right)$. The result follows.

Proposition 40

$\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }=0\beta \mathrm{\pi \pi \pi \pi }=0$

Proposition 41

$\mathrm{\pi \pi \pi \pi }\beta ₯\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }$

$\mathrm{\pi \pi ’\pi \pi \pi \pi \pi \pi \pi }:\mathrm{\pi \pi \pi \pi }\beta ₯2Β·\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }$

$\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }:\mathrm{\pi \pi \pi \pi }\beta ₯\mathrm{\pi \pi \pi }_{\mathrm{\pi \pi \pi \pi }}^{2}$

Proof 41 Similar to PropositionΒ 29.

Proposition 42

$\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }=0\beta \mathrm{\pi \pi \pi \pi \pi \pi \pi }=0$

Proposition 43

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

Proposition 44

$\mathrm{\pi \pi \pi \pi }=0\beta \mathrm{\pi \pi \pi }=0$

Proposition 45

$\mathrm{\pi \pi \pi \pi }\beta ₯\mathrm{\pi \pi \pi }$

Proof 45 Each connected component contains at least one arc (since, by hypothesis, each vertex has at least one arc).

Proposition 46

$\mathrm{\pi \pi \pi \pi }=0\beta \mathrm{\pi \pi \pi \pi }=0$

Proposition 47

$\mathrm{\pi \pi \pi \pi }\beta ₯\mathrm{\pi \pi \pi \pi }$

$\mathrm{\pi \pi }\mathrm{\pi \pi \pi \pi }:\mathrm{\pi \pi \pi \pi }\beta ₯2Β·\mathrm{\pi \pi \pi \pi }$

Proof 47 57 (respectivelyΒ 58) holds since each strongly connected component contains at least one (respectively two) arc(s).

Proposition 48

$\mathrm{\pi \pi \pi \pi }\beta ₯\mathrm{\pi \pi \pi \pi \pi }$

Proof 48 Since isolated vertices are not allowed, each sink has a distinct ingoing arc.

Proposition 49

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

Proof 49 Since isolated vertices are not allowed, each source has a distinct outgoing arc.

Proposition 50

$\mathrm{\pi \pi \pi \pi }=0\beta \mathrm{\pi \pi \pi \pi \pi \pi \pi }=0$

Proposition 51

$\mathrm{\pi \pi \pi \pi }\beta €{\mathrm{\pi \pi \pi \pi \pi \pi \pi }}^{2}$

Proof 51 62 holds since each vertex of a digraph can have at most $\mathrm{\pi \pi \pi \pi \pi \pi \pi }$ successors. The next items correspond to the maximum number of arcs that can be achieved according to a specific arc generator.

Note that, when the equality is reached in 62, the corresponding extreme graph is in fact the graph initially generated. The same observation holds for inequalities 63 to 71. As a consequence all $U$-arcs have to be turned into $T$-arcs.

Proposition 52

$2Β·\mathrm{\pi \pi \pi \pi }\beta ₯\mathrm{\pi \pi \pi \pi \pi \pi \pi }$

Proof 52 By induction on the number of vertices of a graph $G$:

1. If $\mathrm{\pi \pi \pi \pi \pi \pi \pi }\left(G\right)$ is equal to 1 or 2 PropositionΒ 52 holds.

2. Assume that $\mathrm{\pi \pi \pi \pi \pi \pi \pi }\left(G\right)\beta ₯3$.

• Assume there exists a vertex $v$ such that, if we remove $v$, we do not create any isolated vertex in the remaining graph. We have $\mathrm{\pi \pi \pi \pi }\left(G\right)\beta ₯\mathrm{\pi \pi \pi \pi }\left(G-v\right)+1$. Thus $2Β·\mathrm{\pi \pi \pi \pi }\left(G\right)\beta ₯2Β·\mathrm{\pi \pi \pi \pi }\left(G-v\right)+1$. Since by induction hypothesis $2Β·\mathrm{\pi \pi \pi \pi }\left(G-v\right)\beta ₯\mathrm{\pi \pi \pi \pi \pi \pi \pi }\left(G-v\right)=\mathrm{\pi \pi \pi \pi \pi \pi \pi }\left(G\right)-1$ the result holds.

• Otherwise, all the connected components of $G$ are reduced to two elements with only one arc. We remove one of such connected component $\left(v,w\right)$.

Thus $\mathrm{\pi \pi \pi \pi }\left(G\right)=\mathrm{\pi \pi \pi \pi }\left(G-\left\{v,w\right\}\right)+1$. As by induction hypothesis, $2Β·\mathrm{\pi \pi \pi \pi }\left(G-\left\{v,w\right\}\right)\beta ₯\mathrm{\pi \pi \pi \pi \pi \pi \pi }\left(G-\left\{v,w\right\}\right)=\mathrm{\pi \pi \pi \pi \pi \pi \pi }\left(G\right)-2$ the result holds.

Note that, when the equality is reached in 52, the corresponding extreme graph is in fact a perfect matching of the graph. As a consequence all $U$-arcs that do not belong to any perfect matching have to be turned into $F$-arcs.

Proposition 53

Proposition 54

$\mathrm{\pi \pi \pi }=0\beta \mathrm{\pi \pi \pi \pi }=0$

Proposition 55

$\mathrm{\pi \pi \pi }\beta €\mathrm{\pi \pi \pi \pi }$

Proof 55 Holds since each connected component contains at least one strongly connected component.

Note that, when the equality is reached in 55, each connected component of the corresponding extreme graph is strongly connected. As a consequence all sink vertices of the graph induced by the $T$-vertices and the $T$-arcs should have at least one successor.

Proposition 56

$\mathrm{\pi \pi \pi }=0\beta \mathrm{\pi \pi \pi \pi \pi \pi \pi }=0$

Proposition 57

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

$\mathrm{\pi \pi }\mathrm{\pi \pi \pi \pi }:2Β·\mathrm{\pi \pi \pi }\beta €\mathrm{\pi \pi \pi \pi \pi \pi \pi }$

Proof 57 77 (respectivelyΒ 78) holds since each connected component contains at least one (respectively two) vertex.

Note that, when the equality is reached in 77, the corresponding extreme graph does not contain any arc between two distinct vertices. As a consequence any $U$-arc between two distinct vertices is turned into a $F$-vertex.

Proposition 58

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

Proof 58 Holds since between two βconsecutiveβ connected components of the initial graph there is at least one vertex that is missing.

Proposition 59

$\mathrm{\pi \pi \pi \pi }\beta ₯\mathrm{\pi \pi \pi \pi \pi }+1$

Proof 59 Since each sink cannot belong to a circuit and since no isolated vertex is allowed at least one extra non-sink vertex is required the result follows.

Proposition 60

$\mathrm{\pi \pi \pi \pi }\beta ₯\mathrm{\pi \pi \pi \pi \pi \pi \pi }+1$

Proof 60 Since each source cannot belong to a circuit and since no isolated vertex is allowed at least one extra non-source vertex is required the result follows.

Proposition 61

$\mathrm{\pi \pi \pi \pi }=0\beta \mathrm{\pi \pi \pi \pi \pi \pi \pi }=0$

Proposition 62

$\mathrm{\pi \pi \pi \pi }\beta €\mathrm{\pi \pi \pi \pi \pi \pi \pi }$

Proof 62 PropositionΒ 62 holds since each strongly connected component contains at least one vertex.

Proposition 63

$\mathrm{\pi \pi \pi ’\pi \pi \pi \pi }:\mathrm{\pi \pi \pi \pi }=\mathrm{\pi \pi \pi \pi \pi \pi \pi }$

Proof 63 In a directed acyclic graph we have that each vertex corresponds to a strongly connected component involving only that vertex.

Proposition 64

$\mathrm{\pi \pi \pi \pi \pi \pi \pi }=0\beta \mathrm{\pi \pi \pi \pi \pi }=0$

Proposition 65

$\mathrm{\pi \pi \pi \pi \pi \pi \pi }>0\beta \mathrm{\pi \pi \pi \pi \pi }<\mathrm{\pi \pi \pi \pi \pi \pi \pi }$

Proof 65 Holds since each sink must have a predecessor that cannot be a sink and since each vertex has at least one arc.

Proposition 66

$\mathrm{\pi \pi \pi \pi \pi \pi \pi }=0\beta \mathrm{\pi \pi \pi \pi \pi \pi \pi }=0$

Proposition 67

$\mathrm{\pi \pi \pi \pi \pi \pi \pi }>0\beta \mathrm{\pi \pi \pi \pi \pi \pi \pi }<\mathrm{\pi \pi \pi \pi \pi \pi \pi }$

Proof 67 Holds since each source must have a successor that cannot be a source and since each vertex has at least one arc.