Problem 2. Prove the theorem: If there is an xy-walk in a graph G, show that there is also an xy-path in G, using proof by contradiction.

al n+1 #2,2", Q4. (a) Prove that for any two simple connected joint graphs G and H, * L(G) UL(H) L(GUH). Aut (6) iff G contains at

(i) Let e1, e2, . . . , one connection of the graph G and let xi−1 and xi be the sites of the connection ei (1 ≤ i ≤ n). Show that every closed walk (e1, e2, . . . , en) is of length at least 3 with pairs of distinct pointsx1, x2, . . . , xn cycles.(ii) A graph containing no cycles is called acyclic. A walk is acyclic if the subgraph consisting of points and links of the walk is acyclic. Prove: a walk has all distinct points if and only if it is an acyclic sequence.(iii) If and are in different points of the graph G and if there is a walk in G from u to v, show that then there is an acyclic sequence from u to v.(Explain precisely that every shortest walk from u to v is actually an acyclic path.)

1946 4. Give an example to show that if P is a (u, v)-path in a 2-connected graph G, then G does not necessarily contain a (u, v)-path Q internally-disjoint from P.