The super-connectivity of graphs with two orbits

A graph G is said to be super-connected or simply super-κ, if each minimum vertex cut of G isolates a vertex. A graph G is said to be a k-vertex-orbit graph if there are k vertex orbits when Aut(G) acts on V(G). A graph G is said to be a k-edge-orbit graph if there are k edge orbits when Aut(G) acts on edge set E(G). In this paper, we give a necessary and sufficient condition for connected bipartite 2-vertex-orbit graphs to be super-κ. For 2-edge-orbit graphs, we give a sufficient condition for connected 2-edge-orbit graphs to be super-κ. In addition, we show that if G is a k-regular connected irreducible II-kind 2-edge-orbit graph with k ≤ 6 and girth g(G) ≥ 6, or G is a k-regular connected irreducible III-kind 2-edge-orbit graph with k ≤ 6 and girth g(G) ≥ 8, then G is super-connected.