Sparse connectivity certificates via MA orderings in graphs

Abstract

For an undirected multigraph G = ( V , E ) , let α be a positive integer weight function on V. For a positive integer k, G is called ( k , α ) -connected if any two vertices u , v ∈ V remain connected after removal of any pair ( Z , E ′ ) of a vertex subset Z ⊆ V - { u , v } and an edge subset E ′ ⊆ E such that ∑ v ∈ Z α ( v ) + | E ′ | < k . The ( k , α ) -connectivity is an extension of several common generalizations of edge-connectivity and vertex-connectivity. Given a ( k , α ) -connected graph G, we show that a ( k , α ) -connected spanning subgraph of G with O ( k | V | ) edges can be found in linear time by using MA orderings. We also show that properties on removal cycles and preservation of minimum cuts can be extended in the ( k , α ) -connectivity.