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• W2038044706 abstract "This paper examines edge reductions in cyclically k-connected cubic graphs, focusing on when they preserve the cyclic k-connectedness. For a cyclically k-connected cubic graph G, we denote by Nk(G) the set of edges whose reduction gives a cubic graph which is not cyclically k-connected. With the exception of three graphs, Nk(G) consists of the edges in independent k-edge cuts. For this reason we examine the properties and interactions between independent k-edge cuts in cyclically k-connected cubic graphs. These results lead to an understanding of the structure of G[Nk]. For every k, we prove that G[Nk] is a forest with at least k trees if G is a cyclically k-connected cubic graph with girth at least k + 1 and Nk ≠ ⊘. Let fk(ν) be the smallest integer such that |Nk(G)| ≤ fk(ν) for all cyclically k-connected cubic graphs G on ν vertices. For all cyclically 3-connected cubic graphs G such that 6 ≤ ν(G) and N3 ≠ ⊘, we prove that G[N3] is a forest with at least three trees. We determine f3 and state a characterization of the extremal graphs. We define a very restricted subset N4b of N4 and prove that if N4g = N4 − N4b ≠ ⊘, then G[N4g] is a forest with at least four trees. We determine f4 and state a characterization of the extremal graphs. There exist cyclically 5-connected cubic graphs such that E(G) = N5(G), for every ν such that 10 ≤ ν and 16 ≠ ν. We characterize these graphs. Let gk(ν) be the smallest integer such that |Nk(G)| ≤ gk(ν) for all cyclically k-connected cubic graphs G with ν vertices and girth at least k + 1. For k ∈ {3, 4, 5}, we determine gk and state a characterization of the extremal graphs." @default.
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• W2038044706 date "1992-09-01" @default.
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• W2038044706 title "Edge reductions in cyclically k-connected cubic graphs" @default.
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• W2038044706 doi "https://doi.org/10.1016/0095-8956(92)90004-h" @default.
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