The cubic graphs with finite cyclic vertex connectivity larger than girth

Abstract

Cyclic (vertex and edge) connectivity is an important concept in graphs. While cyclic edge connectivity (cλ) has been studied for many years, the study at cyclic vertex connectivity (cκ) is still at the initial stage. And cκ seems to be more complicated than cλ. We have got a sufficient condition that ν(G)≥2g(k−1) for cκ≠∞. On the other hand, if ν(G)<2g(k−1), then we have cκ=∞, or cκ≤(k−2)g, or (k−2)g<cκ<∞. So characterizing all the k-regular graphs with (k−2)g<cκ<∞ is helpful to design an efficient algorithm for cκ. Hence, we characterize all 38 cubic graphs with g<cκ<∞ and prove that cκ=g+1.