G = (V,E), a specified set of vertices S ⊆V with |S|3 and a function g:V→Z+∪{∞}, 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 S and such that vertex-degree increase of each v ∈V by the addition of E' to G is at most g(v), where Z+ is the set of nonnegative integers." This paper shows a linear time algorithm for 2VCA-SV-DC." />

Bi-Connectivity Augmentation for Specified Vertices of a Graph with Upper Bounds on Vertex-Degree Increase

Toshiya MASHIMA  Takanori FUKUOKA  Satoshi TAOKA  Toshimasa WATANABE 

Publication
IEICE TRANSACTIONS on Information and Systems  Vol.E89-D  No.2  pp.751-762
Publication Date: 2006/02/01
Online ISSN: 1745-1361
Print ISSN: 0916-8532
Type of Manuscript: Special Section PAPER (Special Section on Foundations of Computer Science)
Category: Graph Algorithm
Keyword: 
graphs,  vertex-connectivity of a specified set of vertices,  augmentation problems,  degree constraints,  linear time algorithms, 

Full Text: PDF(624.1KB)

Summary: 
The 2-vertex-connectivity augmentation problem for a specified set of vertices of a graph with degree constraints, 2VCA-SV-DC, is defined as follows: "Given an undirected graph G = (V,E), a specified set of vertices S ⊆V with |S|3 and a function g:V→Z+∪{∞}, 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 S and such that vertex-degree increase of each v ∈V by the addition of E' to G is at most g(v), where Z+ is the set of nonnegative integers." This paper shows a linear time algorithm for 2VCA-SV-DC.