On Cyclic Edge-Connectivity and Super-Cyclic Edge-Connectivity of Double-Orbit Graphs

Abstract

A cyclic edge-cut of a graph G is an edge set, the removal of which separates two cycles. If G has a cyclic edge-cut, then it is said to be cyclically separable. For a cyclically separable graph G, the cyclic edge-connectivity \(\lambda _c(G)\) is the cardinality of a minimum cyclic edge-cut of G. Let \(\zeta (G)=\min \{\omega (X) | X \text{ induce } \text{ a } \text{ shortest } \text{ cycle } \text{ in } G\}\), where \(\omega (X)\) is the number of edges with one end in X and the other end in \(V(G)-X\). A cyclically separable graph G with \(\lambda _c(G)=\zeta (G)\) is said to be cyclically optimal. In particular, we call a graph G super cyclically edge-connected if every minimum cyclic edge-cut isolates a shortest cycle of G. In this work, we first discuss the cyclic edge-connectivity of vertex transitive graphs, regular double-orbit graphs, and the double-orbit graphs with two orbits of same size; moreover, we also discuss the super-cyclic edge-connectivity of double-orbit graphs mentioned above.