Generation of cyclically 4-connected cubic graphs Abstract In my talk I prove that the class of cyclically 4-connected cubic graphs can be generated from three small graphs, the complete bipartite graph K_{3,3}, the cube and the twisted cube, by means of two locally defined operations. First one is the reverse of vertex-reduction and the second is 4-reduction. The vertex-reduction of G is defined by removal a vertex v and smoothing the 2-valent vertices of G-v. The 4-reduction removes a 4-edge cut C separating a quadrangle Q, takes H=G-Q and restores the cubicity by adding two edges. The statement can be used in inductive proofs. The core of the proof consists in an argument that one can always find a proper vertex v, or a proper quadrangle Q, so that applying the vertex reduction with respect to v, or 4-reduction with respect to Q, the obtained cubic graph remains cyclically 4-connected.