Edge reductions in cyclically k-connected cubic graphs

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 N_k(G) the set of edges whose reduction gives a cubic graph which is not cyclically k-connected. With the exception of three graphs, N_k(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[N_k]. For every k, we prove that G[N_k] is a forest with at least k trees if G is a cyclically k-connected cubic graph with girth at least k + 1 and $\text{N}{}_{\mathrm{k}}\text{≠ ⊘}$. Let f_k(ν) be the smallest integer such that |N_k(G)| ≤ f_k(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 $\text{N}{}_3\text{≠ ⊘}$, 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 N₄^b of N₄ and prove that if $\text{N}{}_4{}^{\mathrm{g}}\text{= N}{}_4\text{− N}{}_4{}^{\mathrm{b}}\text{≠ ⊘}$, then G[N₄^g] 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 g_k(ν) be the smallest integer such that |N_k(G)| ≤ g_k(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 g_k and state a characterization of the extremal graphs.