Exercises $1\mathrm{.}6$

I$1.$Implement a weak sequential $($in any order$)$ deletion of vertices of $C_4,W_5,K_5,K_{3,3}$ and Petersen graph.

$2.$Implement a strong sequential $($in any order$)$ deletion of vertices of $C_4,W_5,K_5,K_{3,3}$ and Petersen graph.

$3.$ Implement a weak sequential $($in any order$)$ deletion of edges of $C_4,W_5,K_5,K_{3,3}$ and Petersen graph.

$4.$ Implement a strong sequential $($in any order$)$ deletion of edges of $C_4,W_5,K_5,K_{3,3}$ and Petersen graph; at each step, weakly remove the loops.

$5.$ mplement a sequential $($in any order$)$ contraction of edges of $C_4,W_5,K_5,K_{3,3}$ and Petersen graph; at each step, weakly remove multiple edges.

$6.$ Find the Hadwiger number of any tree, $C_5,W_5,K_{2,3}.$

$7.$ Construct the complement of $C_3,C_4,C_5,C_6,K_5,K_{3,5},$ Petersen graph, cube, $P_7,$ any tree on $6$ vertices, $E_5,2C_3.$

$8.$ Show that $\frac{?}{\text{b}\text{a}\text{r}}(C{)}_5$~$=C_5.$

$9.$ Show that $\frac{?}{\text{b}\text{a}\text{r}}(C{)}_6$ is isomorphic to a prism.

$10.$ How are the degree sequences of $G$ and $\frac{?}{\text{b}\text{a}\text{r}}(G)$ related?