Connectivity of the graph induced by contractible edges of a k-tree

Abstract

A k-tree is a Kk+1 or a graph on at least k+2 vertices obtained from a smaller k-tree by adding one vertex and joining it to the vertices of a k-clique. Let G be a k-connected graph, and let e be an edge of G. The edge e is said to be contractible if the graph obtained from G by contracting e is again a k-connected graph, otherwise it is said to be non-contractible. Let G be a k-tree, and let Gc=(V(G),EC(G)), where EC(G) denotes the set of all contractible edges of G. In this paper, we prove that κ(Gc)=δ(Gc). Further, Gc is super connected, whenever 3 ≤ δ(Gc) < k.