G = (V,E) and an upper bound a(v;G) Z+{} on vertex-degree increase for each v V, find a smallest set E′ of edges such that (V,E E′) has at least two internally-disjoint paths between any pair of vertices in V and such that vertex-degree increase of each v V by the addition of E′ to G is at most a(v;G), where Z+ is the set of nonnegative integers." In this paper we show that checking the existence of a feasible solution and finding an optimum solution to 2VCA-DC can be done in O(|V|+|E|) time." />

A Linear Time Algorithm for Bi-Connectivity Augmentation of Graphs with Upper Bounds on Vertex-Degree Increase

Takanori FUKUOKA
Toshiya MASHIMA
Satoshi TAOKA
Toshimasa WATANABE

Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E88-A    No.4    pp.954-963
Publication Date: 2005/04/01
Online ISSN: 
DOI: 10.1093/ietfec/e88-a.4.954
Print ISSN: 0916-8508
Type of Manuscript: Special Section PAPER (Special Section on Selected Papers from the 17th Workshop on Circuits and Systems in Karuizawa)
Category: 
Keyword: 
graphs,  connectivity augmentation,  vertex-connectivity,  degree constraints,  linear time algorithms,  

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Summary: 
The 2-vertex-connectivity augmentation problem of a graph with degree constraints, 2VCA-DC, is defined as follows: "Given an undirected graph G = (V,E) and an upper bound a(v;G) Z+{} on vertex-degree increase for each v V, find a smallest set E′ of edges such that (V,E E′) has at least two internally-disjoint paths between any pair of vertices in V and such that vertex-degree increase of each v V by the addition of E′ to G is at most a(v;G), where Z+ is the set of nonnegative integers." In this paper we show that checking the existence of a feasible solution and finding an optimum solution to 2VCA-DC can be done in O(|V|+|E|) time.

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