Prove that connecting two nodes u and v in a graph G by a new edge creates a new cycle if and only if u and v are in the same connected component of G.
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Prove that connecting two nodes u and v in a graph G by a new edge creates a new cycle if and only if u and v are in the same connected component of G.
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- Prove that for any two edges of a 2-connected graph, a cycle exists containing both of them.Let Vn be the set of connected graphs having n edges, vertex set [n], and exactly one cycle. Form a graph Gn whose vertex set is Vn. Include {gn, hn} as an edge of Gn if and only if gn and hn differ by two edges, i.e. you can obtain one from the other by moving a single edge. Tell us anything you can about the graph Gn. For example, (a) How many vertices does it have? (b) Is it regular (i.e. all vertices the same degree)? (c) Is it connected? (d) What is its diameter?Show that if an edge e is in a closed, trail of G, then e is in a cycle of G
- True or False: The edge set of every closed trail in a graph can be partitioned into edge sets of cycles2. Let G be a graph and e € E(G). Let H be the graph with V(H) = V(G) and E(H) = E(G)\{e}. Then e is a bridge of G if H has a greater number of connected components than G. (b) Assume that G is connected and that e is a bridge of G with endpoints u and v. Show that H has exactly two connected components H₁ and H₂ with u € V(H₁) and v € V (H₂). To this end, you may want to consider an arbitrary vertex w € V (G) and use a u-w-path in G to construct a u-w-path or a v-w-path in H.Let G be a graph and e € E(G). Let H be the graph with V(H) = V(G) and E(H) = E(G)\ {e}. Then e is a bridge of G if H has a greater number of connected components than G. Assume that G is connected and that e is a bridge of G with endpoints u and v. Show that H has exactly two connected components H₁ and H₂ with u € V (H₁) and v € V(H₂). To this end, you may want to consider an arbitrary vertex w ¤ V (G) and use a u-w-path in G to construct a u-w-path or a v-w-path in H.
- An edge is called a bridge if the removal of the edge increases the number of connected components in G by one. The removal of a bridge thus separates a component of G into two separate components. Let G be a graph on 6 vertices and 8 edges. How many bridges can G have at the most?Draw the directed graphs of the relation.Need to make this graph in R
- Let G be a graph and e € E(G). Let H be the graph with V(H) = V(G) and E(H) = E(G)\{e}. Then e is a bridge of G if H has a greater number of connected components than G. Show that e is a bridge of G if and only if it is not contained in a cycle of G.Prove that a simple 2-connected graph G with at least four vertices is 3-connected if and only if for every triple (x, y, z) of distinct vertices and any edge e not incident with y, G has an x, z-path through e that does not contain y.Let u and v be distinct vertices in a connected graph G. There may be several connected subgraphs of G containing u and v. What is the minimum size of a connected subgraph of G containing u and v? Explain your answer.