\(V = \{A, B, C, D, E, F, G, H, I\}\) \(E = \{e1 = (A, B), e2 = (A, C), e3 = (A, D), e4 = (B, C), e5 = (B, F), e6 = (B, H), e7 = (C, D), e8 = (C, E), e9 = (C, F), e10 = (D, E), e11 = (D, I), e12 = (E, F), e13 = (E, G), e14 = (F, G), e15 = (F, H), e16 = (G, H), e17 = (G, I), e18 = (H, I)\}\) \(W = \{22, 9, 12, 35, 36, 34, 4, 33, 30, 18, 23, 24, 39, 25, 21, 19\}\) ....................... is a cut-set in the above graph.
\(\{e1, e4, e5, e6\}\)
Definition: a cut-set is a minimal set of edges whose removal disconnects a connected graph.
Check \(\{e1, e2, e4\}\): A still reaches D through e3 and B still reaches F and H through e5, e6, so the graph stays connected.
Check \(\{e12, e13, e15, e16\}\): E keeps e8, e10, G keeps e14, e17 and H keeps e6, e18, so the graph stays connected.
Check \(\{e4, e7, e8, e9\}\): C keeps e2 to A, and A, B, D, E, F are linked by the remaining edges, so the graph stays connected.
Edges incident on B: \(e1 = (A, B), e4 = (B, C), e5 = (B, F), e6 = (B, H)\). Removing all four isolates B, so the graph splits into \(\{B\}\) and the other eight vertices.
Minimality: putting back any one of these edges reconnects B, so no proper subset disconnects the graph.
Hence, \(\{e1, e4, e5, e6\}\).
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