Sparse connectivity certificates via MA orderings in graphs

For an undirected multigraph G=(V,E)G=(V,E), let αα be a positive integer weight function on V. For a positive integer k, G   is called (k,α)(k,α)-connected if any two vertices u,v∈Vu,v∈V remain connected after removal of any pair (Z,E′)(Z,E′) of a vertex subset Z⊆V-{u,v}Z⊆V-{u,v} and an edge subset E′⊆EE′⊆E such that ∑v∈Zα(v)+|E′|