Sparse Spanning Subgraphs Preserving Connectivity and Distance between Vertices and Vertex Subsets

Hiroyoshi MIWA  Hiro ITO  

Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E81-A    No.5    pp.832-841
Publication Date: 1998/05/25
Online ISSN: 
DOI: 
Print ISSN: 0916-8508
Type of Manuscript: Special Section PAPER (Special Section on Discrete Mathematics and Its Applications)
Category: 
Keyword: 
graph,  area graph,  diameter,  distance,  connectivity,  NP-complete,  polynomial time,  algorithm,  

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Summary: 
This paper investigates the relations between the computational complexity and the restrictions for several problems that determine whether a given graph with edge costs and edge lengths has a spanning subgraph with such restrictions as the diameter, the connectivity, and the NA-distance and the NA-(edge)-connectivity proposed and investigated in [1]-[5]. The NA-distance and the NA-(edge)-connectivity are the measures for the distance and the connectivity between a vertex and a vertex subset (area). In this paper we prove that the minimum diameter spanning subgraph problem considering the restrictions of the diameter and the sum of edge costs is NP-complete even if the following restrictions are satisfied: all edge costs and all edge lengths are equal to one, and the upper bound of the diameter is restricted to four. Next, we prove that the minimum NA-distance spanning subgraph problem considering the restrictions of the NA-distances and the sum of edge costs is NP-complete even if the following conditions are satisfied: all edge costs and all edge lengths are equal to one, the upper bound of the NA-distance is restricted to four, each area is composed of a vertex, and the number of areas is restricted to two. Finally, we investigate the preserving NA-distance and NA-edge-connectivity spanning subgraph problem considering the preservations of the NA-distances and the NA-edge-connectivity and the restrictions of the sum of edge costs, and prove that a sparse spanning subgraph can be constructed in polynomial time if all edge costs are equal to one.