Making the components of a graph k-connected

V. Nikiforov, R.H. Schelp
2007 Discrete Applied Mathematics  
For every integer k 2 and graph G, consider the following natural procedure: if G has a component G that is not k-connected, remove G if |G | k, otherwise remove a cutset U ⊂ V (G ) with |U | < k; do the same with the remaining graph until only k-connected components are left or all vertices are removed. We are interested when this procedure stops after removing o(|G|) vertices. Surprisingly, for every graph G of order n with minimum degree (G) √ 2(k − 1)n, the procedure always stops after
more » ... ing at most 2n(k − 1)/ vertices. We give examples showing that our bounds are essentially best possible.
doi:10.1016/j.dam.2006.07.007 fatcat:ow6lukeq4rdmjavlrohr4tnu4m