Suppose that G = (V, E) is a directed graph. A vertex w E V is called reachable from a vertex v € V if there is a directed path from v to w. The vertices v and w are mutually reachable if there is both a directed path from v to w and a directed path from w to v in G. Show that if G = (V, E) is a directed graph and that u, v and w are vertices in V for which u and v are mutually reachable and v and w are mutually reachable, then u and w are mutually reachable.

I have to prove the following Corollary: "Let u and v be vertices of a 2-connected graph G. Then there is a cycle of G that contains both u and v.”   My idea of the proof is the following: Given vertices u and v, I want to show that there is a cycle containing both. Then by contradiction, I could assume that there exists u and v sucht that there are no cycles C including u and v.  However, I'm not quite sure that this is the correct approach. I would appreciate some help to prove this corollary. Thank youuu

Construct a graph with vertices U, V, W, X,Y that has an Euler path, the degree of U is 1 and the degree of Y is 3. What is the edge set?

Help me  with b and c

-nsider the following graph G(V, E) w • The set of vertices V: {2,3, 4, 6, 8, • The set of edges E: {(a, b) e Z² : « a) Draw the graph G. b) Draw the graph G3

how can I prove that "if G has no isolated vertices and has a walk that uses all the edges, then G is connected."