Results 1  10
of
41
Property Testing in Bounded Degree Graphs
 Algorithmica
, 1997
"... We further develop the study of testing graph properties as initiated by Goldreich, Goldwasser and Ron. Whereas they view graphs as represented by their adjacency matrix and measure distance between graphs as a fraction of all possible vertex pairs, we view graphs as represented by boundedlength in ..."
Abstract

Cited by 120 (36 self)
 Add to MetaCart
We further develop the study of testing graph properties as initiated by Goldreich, Goldwasser and Ron. Whereas they view graphs as represented by their adjacency matrix and measure distance between graphs as a fraction of all possible vertex pairs, we view graphs as represented by boundedlength incidence lists and measure distance between graphs as a fraction of the maximum possible number of edges. Thus, while the previous model is most appropriate for the study of dense graphs, our model is most appropriate for the study of boundeddegree graphs. In particular, we present randomized algorithms for testing whether an unknown boundeddegree graph is connected, kconnected (for k ? 1), planar, etc. Our algorithms work in time polynomial in 1=ffl, always accept the graph when it has the tested property, and reject with high probability if the graph is fflaway from having the property. For example, the 2Connectivity algorithm rejects (w.h.p.) any Nvertex ddegree graph for which more ...
Biconnectivity Approximations and Graph Carvings
, 1994
"... A spanning tree in a graph is the smallest connected spanning subgraph. Given a graph, how does one find the smallest (i.e., least number of edges) 2connected spanning subgraph (connectivity refers to both edge and vertex connectivity, if not specified) ? Unfortunately, the problem is known to be ..."
Abstract

Cited by 82 (3 self)
 Add to MetaCart
A spanning tree in a graph is the smallest connected spanning subgraph. Given a graph, how does one find the smallest (i.e., least number of edges) 2connected spanning subgraph (connectivity refers to both edge and vertex connectivity, if not specified) ? Unfortunately, the problem is known to be NP hard. We consider the problem of finding a better approximation to the smallest 2connected subgraph, by an efficient algorithm. For 2edge connectivity our algorithm guarantees a solution that is no more than 3 2 times the optimal. For 2vertex connectivity our algorithm guarantees a solution that is no more than 5 3 times the optimal. The previous best approximation factor is 2 for each of these problems. The new algorithms (and their analyses) depend upon a structure called a carving of a graph, which is of independent interest. We show that approximating the optimal solution to within an additive constant is NP hard as well. We also consider the case where the graph has edge weigh...
Approximating minimum cost connectivity problems
 58 in Approximation algorithms and Metaheuristics, Editor
, 2007
"... ..."
Minimal EdgeCoverings of Pairs of Sets
, 1995
"... A new minmax theorem concerning bisupermodular functions on pairs of sets is proved. As a special case, we derive an extension of (A. Lubiw's extension of) E. Györi's theorem on intervals, W. Mader's theorem on splitting off edges in directed graphs, J. Edmonds' theorem on matroid partitions, and ..."
Abstract

Cited by 57 (13 self)
 Add to MetaCart
A new minmax theorem concerning bisupermodular functions on pairs of sets is proved. As a special case, we derive an extension of (A. Lubiw's extension of) E. Györi's theorem on intervals, W. Mader's theorem on splitting off edges in directed graphs, J. Edmonds' theorem on matroid partitions, and an earlier result of the first author on the minimum number of new directed edges whose addition makes a digraph kedgeconnected. As another consequence, we solve the corresponding nodeconnectivity augmentation problem in directed graphs.
Preserving And Increasing Local EdgeConnectivity In Mixed Graphs
 SIAM J. Discrete Math
, 1995
"... Generalizing and unifying earlier results of W. Mader and of the second and third authors, we prove two splitting theorems concerning mixed graphs. By invoking these theorems we obtain minmax formulae for the minimum number of new edges to be added to a mixed graph so that the resulting graph satis ..."
Abstract

Cited by 22 (7 self)
 Add to MetaCart
Generalizing and unifying earlier results of W. Mader and of the second and third authors, we prove two splitting theorems concerning mixed graphs. By invoking these theorems we obtain minmax formulae for the minimum number of new edges to be added to a mixed graph so that the resulting graph satisfies local edgeconnectivity prescriptions. An extension of Edmonds' theorem on disjoint arborescences is also deduced along with a new sufficient condition for the solvability of the edgedisjoint paths problem in digraphs. The approach gives rise to strongly polynomial algorithms for the corresponding optimization problems. 1. INTRODUCTION AND PRELIMINARIES Our main concern, the edgeconnectivity augmentation problem, is as follows. What is the minimum number (or, more generally, the minimum cost) fl of new edges to be added to M so that in the resulting graph M 0 the local edgeconnectivity (x; y; M 0 ) between every pair of nodes x; y is at least a prescribed value r(x; y)? Several ...
Edgeconnectivity augmentation with partition constraints
 SIAM J. Discrete Mathematics
, 1999
"... When k is even the minmax formula for the partitionconstrained problem is a natural generalization of [3]. However this generalization fails when k is odd. We show that at most one more edge is needed when k is odd and we characterize the graphs that require such an extra edge. ..."
Abstract

Cited by 17 (9 self)
 Add to MetaCart
When k is even the minmax formula for the partitionconstrained problem is a natural generalization of [3]. However this generalization fails when k is odd. We show that at most one more edge is needed when k is odd and we characterize the graphs that require such an extra edge.
A Linear Time Algorithm for Triconnectivity Augmentation (Extended Abstract)
 IN PROC. 32TH ANNUAL IEEE SYMP. ON FOUNDATIONS OF COMP. SCI
, 1991
"... We consider the problem of finding a smallest set of edges whose addition triconnects an undirected graph. This is a fundamental graphtheoretic problem that has applications in designing reliable networks and faulttolerant computing. We present a linear time sequential algorithm for the problem. Th ..."
Abstract

Cited by 16 (5 self)
 Add to MetaCart
We consider the problem of finding a smallest set of edges whose addition triconnects an undirected graph. This is a fundamental graphtheoretic problem that has applications in designing reliable networks and faulttolerant computing. We present a linear time sequential algorithm for the problem. This is a substantial improvement over the best previous algorithm for this problem, which runs in O(n(n+m)²) time on a graph with n vertices and m edges.
Covering Symmetric Supermodular Functions by Graphs
, 1998
"... The minimum number of edges of an undirected graph covering a symmetric, supermodular setfunction is determined. As a special case, we derive an extension of a theorem of J. BangJensen and B. Jackson on hypergraph connectivity augmentation. ..."
Abstract

Cited by 15 (3 self)
 Add to MetaCart
The minimum number of edges of an undirected graph covering a symmetric, supermodular setfunction is determined. As a special case, we derive an extension of a theorem of J. BangJensen and B. Jackson on hypergraph connectivity augmentation.