Some conditional vertex connectivities of complete-transposition graphs

A subset F⊂V(G)F⊂V(G) is called an RkRk-vertex-cut if G-FG-F is disconnected and each vertex u∈V(G)-Fu∈V(G)-F has at least k   neighbors in G-FG-F. The cardinality of a minimum RkRk-vertex-cut is the RkRk-vertex-connectivity of G   and is denoted by κk(G)κk(G). The conditional connectivity is a measure to study the structure of networks beyond connectivity. Hypercubes form the basic classes of interconnection networks. Complete transposition graphs were introduced to be competitive models of hypercubes. In this paper, we determine the numbers κ1κ1 and κ2κ2 for complete-transposition graphs, κ1(CTn)=n(n-1)-2,κ2(CT4)=16 and κ2(CTn)=2n(n-1)-10κ2(CTn)=2n(n-1)-10 for n⩾5n⩾5.