1. Can a graph has a cut-vertex of degree one? Explain the reason for your answer. 2. Prove that: if S is a set of vertices in G, then 1[ 5,S ] = vES deg(v) — 2 e(G[S]). 3. Let G be a simple connected...

1. Can a graph has a cut-vertex of degree one? Explain the reason for your answer.
2. Prove that: if S is a set of vertices in G, then 1[ 5,S ] =
vES
deg(v) — 2 e(G[S]).
3. Let G be a simple connected graph with blocks B1, B2, Bt. Prove that G has exactly
)(Bi) — t + 1 vertices. (Hint: Use induction on the number of blocks t)
4. Let G be a connected graph with an edge cut F. Prove that if G — F has exactly two components then the edge cut F is a bond. 5. Let G is a simple graph with A 3. Show that ki(G) = K(G).