On cyclically 4-connected cubic graphs

A 3-connected cubic graph is cyclically 4-connected if it has at least n≥8 vertices and when removal of a set of three edges results in a disconnected graph, only one component has cycles. By introducing the notion of cycle spread to quantify the distance between pairs of edges, we get a new characterization of cyclically 4-connected graphs. Let Qn and Vn denote the ladder and Möbius ladder on n≥8 vertices, respectively. We prove that a 3-connected cubic graph G is cyclically 4-connected if and only if G is either the Petersen graph, Qn or Vn for n≥8, or G is obtained from Q8 or Q10 by bridging pairs of edges with cycle spread at least (1,2). The concept of cycle spread also naturally leads to methods for constructing cyclically k-connected cubic graphs from smaller ones, but for k≥5 the method is not exhaustive.