TAOCP 2.3.4.1 Exercise 7
Let the graph of the flow chart have vertices corresponding to the boxes, and let the free subtree be $G'$ with edges $e_1,e_2,e_3,e_4,e_9$.
Exercise 7. [**] $$22$$ Carry out the construction in the text for the flow chart
using the free subtree consisting of edges $e_1,e_2,e_3,e_4,e_9$. What are the fundamental cycles? Express $E_1,E_2,E_3,E_4,E_9$ in terms of $E_5,E_6,E_7$, and $E_8$.
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Let the graph of the flow chart have vertices corresponding to the boxes, and let the free subtree be $G'$ with edges $e_1,e_2,e_3,e_4,e_9$. The edges not in $G'$ are $e_5,e_6,e_7,e_8$. For each such edge, adding it to $G'$ produces a unique fundamental cycle by Theorem A. Adding $e_5$ to $G'$ gives the cycle $C_5=(e_5,e_2,e_1)$, adding $e_6$ gives $C_6=(e_6,e_3,e_2)$, adding $e_7$ gives $C_7=(e_7,e_4,e_3)$, and adding $e_8$ gives $C_8=(e_8,e_9,e_4)$. These are all the fundamental cycles.
The independent variables are the $E$-values on edges not in $G'$, namely $E_5,E_6,E_7,E_8$. Using Kirchhoff's law (1) on each vertex and the uniqueness of solutions for the $E$-values on $G'$, the dependent variables are expressed as
$$ E_1 \leftarrow -E_5, \quad E_2 \leftarrow -E_6 + E_5, \quad E_3 \leftarrow -E_7 + E_6, \quad E_4 \leftarrow -E_8 + E_7, \quad E_9 \leftarrow -E_8. $$
Each replacement corresponds to computing the $E$-value of one edge from the independent $E$-values. The total number of replacement operations is $5$.
This completes the construction. ∎
The fundamental cycles are
$$ \begin{aligned} C_5 &: \quad e_5+e_2+e_1,\ C_6 &: \quad e_6+e_3+e_2,\ C_7 &: \quad e_7+e_4+e_3,\ C_8 &: \quad e_8+e_9+e_4. \end{aligned} $$
The dependent variables $E_1,E_2,E_3,E_4,E_9$ are uniquely determined by $E_5,E_6,E_7,E_8$ as given above.