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The number of removable edges in 3-connected graphs. (English) Zbl 0931.05044
Summary: An edge of a 3-connected graph $$G$$ is said to be removable if $$G-e$$ is a subdivision of a 3-connected graph. D. A. Holton, B. Jackson, A. Saito and N. C. Wormald [J. Graph Theory 14, No. 4, 465-473 (1990; Zbl 0729.05037)] proved that every 3-connected graph of order at least five has at least $$\lceil(|G|+ 10)/6\rceil$$ removable edges. We prove that every 3-connected graph of order at least five, except the wheels $$W_5$$ and $$W_6$$, has at least $$(3|G|+ 18)/7$$ removable edges. We also characterize the graphs with $$(3|G|+ 18)/7$$ removable edges. $$\copyright$$ Academic Press.
Reviewer: Reviewer (Berlin)

##### MSC:
 05C40 Connectivity 05C75 Structural characterization of families of graphs
##### Keywords:
3-connected graph; subdivision; removable edges
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##### References:
 [1] Bondy, J.A.; Murty, U.S.R., Graph theory with application, (1981), North-Holland New York · Zbl 1134.05001 [2] Fouquet, J.L.; Thuiller, H., K-minimal 3-connected cubic graphs, Ars combin., 26, 149-190, (1988) · Zbl 0704.05025 [3] Holton, D.A.; Jackson, B.; Saito, A.; Wormald, N.C., Removable edges in 3-connected graphs, J. graph theory, 14, 465-475, (1990) · Zbl 0729.05037 [4] McCuaig, W., Edge reductions in cyclically k-connected cubic graphs, J. combin. theory ser. B, 56, 16-44, (1992) · Zbl 0711.05030
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