Edge-elimination is an operation of removing an edge of a cubic graph together with its endvertices and suppressing the resulting 2-valent vertices. We study the effect of this operation on the cyclic connectivity of a cubic graph. Disregarding a small number of cubic graphs with no more than six vertices, this operation cannot decrease cyclic connectivity by more than two. We show that apart from three exceptional graphs (the cube, the twisted cube, and the Petersen graph) every 2-connected cubic graph on at least eight vertices contains an edge whose elimination decreases cyclic connectivity by at most one. The proof reveals an unexpected behaviour of connectivity 6, which requires a detailed structural analysis featuring the Isaacs flower snarks and their natural generalisation, the twisted Isaacs graphs, as forced structures. A complete characterisation of this family, which includes the Heawood graph as a sporadic case, serves as the main tool for excluding the existence of exceptional graphs in connectivity 6. As an application we show that every cyclically 5-edge-connected cubic graph has a decycling set of vertices whose removal leaves a tree and the set itself has at most one edge between its vertices. This strengthens a classical result of Payan and Sakarovitch (1975) about the structure of minimum decycling sets in cyclically 4-edge-connected graphs.

边消除是一种将三次图的边连同其端顶点一起删除并抑制生成的 2 价顶点的操作。我们研究了该操作对三次图的循环连通性的影响。忽略不超过六个顶点的少量三次图，此操作不能将循环连通性减少两个以上。我们表明，除了三个特殊图（立方体、扭曲立方体和彼得森图）之外，每个至少有八个顶点上的 2 连通三次图都包含一条边，其消除最多会使循环连通性减少一个。证明揭示了连通性 6 的意外行为，这需要详细的结构分析，将 Isaacs 花蛇及其自然概括，扭曲的 Isaacs 图作为强制结构。该族的完整表征，包括作为零星案例的 Heawood 图，作为排除连通性 6 中异常图存在的主要工具。作为一个应用程序，我们展示了每个循环 5 边连通的三次图都有一个回收一组顶点，其移除会留下一棵树，并且该集合本身在其顶点之间最多有一条边。这加强了 Payan 和 Sakarovitch (1975) 关于循环 4 边连接图中最小循环集结构的经典结果。作为一个应用程序，我们展示了每个循环 5 边连接的三次图都有一个循环的顶点集，这些顶点的移除会留下一棵树，并且该集合本身在其顶点之间最多有一条边。这加强了 Payan 和 Sakarovitch (1975) 关于循环 4 边连接图中最小循环集结构的经典结果。作为一个应用程序，我们展示了每个循环 5 边连接的三次图都有一个循环的顶点集，这些顶点的移除会留下一棵树，并且该集合本身在其顶点之间最多有一条边。这加强了 Payan 和 Sakarovitch (1975) 关于循环 4 边连接图中最小循环集结构的经典结果。