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Algebra & Trigonometry with Analytic Geometry

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter3: Functions And Graphs

Section3.3: Lines

Problem 14E

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Let G = (V, E) be a graph and let e be an edge of G. Remember that G\e is produced by
deleting e from G; in other words G\e = (V,E − {e}).
Prove that if G\e is connected, then e is contained in a cycle of G.

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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.
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.
Which of the following graphs contain u,v, and u*v?
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.
Let G be a graph and e 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. Let G be the simple graph with V (G) = {u, v, w, x, y, z) and E(G) = {uy, vx, vz, wx, xz}. For each e E E(G), state whether e is a bridge of G. Justify your answer.
Consider the graph G with • V(G) = {2,3,6} • e(G) = {a, b, c, d, e, f, g} •E(G) = {(a, [2,2]), (b, [3,3]), (c, [6,6]), (d, [2,6]), (e, [6,2]), (f, [3,6]), (g, [6,3])} Using edge connectivity, we can define the relation R = {(2, 2), (3, 3), (6, 6), (2, 6), (6, 2), (3, 6), (6,3)} Which of the following statements are true? [More than one statement may be true.] R is reflexive. R is symmetric. R is transitive. R is antisymmetric.
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?
Use the graph of y = Ca* to determine C and a. C= 10- 8 6- -10-8 -6-4-2 -4- -6- -8- 10- [-10, 10] [-10, 10] Xscl=1 Yscl=1 y 6
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. = (a) Let G be the simple graph with V(G) {u, v, w, x, y, z) and E(G) {uy, vx, vz, wx, xz}. For each e € E(G), state whethere is a bridge of G. Justify your answer.
2. 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.
Show that if an edge e is in a closed, trail of G, then e is in a cycle of G

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