A Kind of Conditional Connectivity of Cayley Graphs Generated by 2-trees

For a connected graph G=(V(G),E(G))⁠, a subset F⊂V(G) is called an Rk-vertex-cut if G−F is disconnected and each vertex u∈V(G)−F has at least k neighbors in G−F⁠. The cardinality of a minimum Rk-vertex-cut of G is the Rk-vertex-connectivity and is denoted by κk(G)⁠. The conditional connectivity is a new measure to study the fault tolerance of network structures beyond connectivity. In this paper, we study R1-vertex-connectivity and R2-vertex-connectivity of Cayley graphs generated by 2-trees T2,n⁠, which are denoted by KTn⁠, and show that κ1(KTn)=4n−8 for n≥4⁠; κ2(KTn)=8n−22 for n≥6⁠.