Sparse connectivity certificates via MA orderings in graphs

Authors:
Hiroshi Nagamochi
Affiliations:
Department of Applied Mathematics and Physics, Kyoto University, Yoshida Honmachi, Sakyo, Kyoto, Japan
Venue:
Discrete Applied Mathematics - Special issue: Discrete algorithms and optimization, in honor of professor Toshihide Ibaraki at his retirement from Kyoto University
Year:
2006

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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 Σv∈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.