Answered: Let G be a connected graph with at…

Let G be a connected graph with at least one edge and F ⊆ E(G) be an edge cut. Prove that F is a minimal edge cut if and only if G − F contains exactly two connected components

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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Let G be a connected graph with at least one edge and F ⊆ E(G) be an edge cut. Prove that F is a minimal edge cut if and only if G − F contains exactly two connected components.

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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.