Exercises $1\mathrm{.}8.$

For every pair of disjoint vertices of graph $G,$ see Figure $1\mathrm{.}36,$ find a separator.

For every pair of disjoint vertices of graph $G,$ see Figure $1\mathrm{.}36,$ find a minimal separator.

For graph $G$ in Figure $1\mathrm{.}36,$ find the connectivity $(G).$

For which integer $k0,$ graph $G$ in Figure $1\mathrm{.}36$ is $k-$connected?

For separator $\{1,3,4,7,8\},$ construct the derived subgraphs.

For separator $\{1,8,3,4,6\},$ construct the derived subgraphs.

For separator $\{5,6,7,8,2\},$ construct the derived subgraphs

Apply Proposition $1\mathrm{.}8\mathrm{.}1$ for separator $\{1,8,3\}.$

Find an edge$-$separator for graph $G$ in Figure $1\mathrm{.}36.$

For graph $G$ in Figure $1\mathrm{.}36$ find a minimal $(1,4)-$separator.

For graph $G$ in Figure $1\mathrm{.}36$ find a minimal $(1,4)-$edge$-$separator.

What is the smallest number of edges that disconnect graph $G$ in Figure $1\mathrm{.}36?$

Apply Menger's Theorem to vertices $3$ and $6$ in graph $G,$ see Figure $1\mathrm{.}36.$