= 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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Transcribed Image Text:=
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.
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