= 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. (a) Let G be the simple graph with V(G) = {u, v, w, x, y, z) and E(G) z} {uy, vx, vz, wx, xz}. For each e € E(G), state whether e is a bridge of G. Justify your answer. = (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 wE V (G) and use a u-w-path in G to construct a u-w-path or a v-w-path in H. (c) 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. (a) Let G be the simple graph with V(G) = {u, v, w, x, y, z) and E(G) z} {uy, vx, vz, wx, xz}. For each e € E(G), state whether e is a bridge of G. Justify your answer. = (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 wE V (G) and use a u-w-path in G to construct a u-w-path or a v-w-path in H. (c) Show that e is a bridge of G if and only if it is not contained in a cycle of G.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter9: Systems Of Equations And Inequalities

Section9.3: Systems Of Inequalities

Problem 30E

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