Theory of linear and integer programming
Theory of linear and integer programming
A polynomial-time algorithm to find the shortest cycle basis of a graph
SIAM Journal on Computing
Journal of the ACM (JACM)
Algorithms for Generating Fundamental Cycles in a Graph
ACM Transactions on Mathematical Software (TOMS)
Tree spanners in planar graphs
Discrete Applied Mathematics - Special issue on international workshop of graph-theoretic concepts in computer science WG'98 conference selected papers
A Polynomial Time Algorithm to Find the Minimum Cycle Basis of a Regular Matroid
SWAT '02 Proceedings of the 8th Scandinavian Workshop on Algorithm Theory
Minimum Cycle Bases for Network Graphs
Algorithmica
A greedy approach to compute a minimum cycle basis of a directed graph
Information Processing Letters
A polynomial time algorithm for minimum cycle basis in directed graphs
STACS'05 Proceedings of the 22nd annual conference on Theoretical Aspects of Computer Science
On a Special Co-cycle Basis of Graphs
SWAT '08 Proceedings of the 11th Scandinavian workshop on Algorithm Theory
An improved heuristic for computing short integral cycle bases
Journal of Experimental Algorithmics (JEA)
Fine-grained boundary recognition in wireless ad hoc and sensor networks by topological methods
Proceedings of the tenth ACM international symposium on Mobile ad hoc networking and computing
A note on fundamental, non-fundamental, and robust cycle bases
Discrete Applied Mathematics
Minimum Cycle Bases and Their Applications
Algorithmics of Large and Complex Networks
Minimum Cycle Bases of Weighted Outerplanar Graphs
ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
Novel topological framework for adaptive routing
GLOBECOM'09 Proceedings of the 28th IEEE conference on Global telecommunications
Minimum cycle bases of weighted outerplanar graphs
Information Processing Letters
On the approximability of the minimum strictly fundamental cycle basis problem
Discrete Applied Mathematics
Cyclic matrices of weighted digraphs
Discrete Applied Mathematics
Properties of Gomory-Hu co-cycle bases
Theoretical Computer Science
Efficient deterministic algorithms for finding a minimum cycle basis in undirected graphs
IPCO'10 Proceedings of the 14th international conference on Integer Programming and Combinatorial Optimization
Survey: Cycle bases in graphs characterization, algorithms, complexity, and applications
Computer Science Review
Integral cycle bases for cyclic timetabling
Discrete Optimization
Minimum cycle bases in graphs algorithms and applications
MFCS'07 Proceedings of the 32nd international conference on Mathematical Foundations of Computer Science
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In the last years, new variants of the minimum cycle basis (MCB) problem and new classes of cycle bases have been introduced, as motivated by several applications from disparate areas of scientific and technological inquiries. At present, the complexity status of the MCB problem has been settled only for undirected, directed, and strictly fundamental cycle bases. In this paper, we offer an unitary classification accommodating these three classes and further including the following four relevant classes: 2-bases (or planar bases), weakly fundamental cycle bases, totally unimodular cycle bases, and integral cycle bases. The classification is complete in that, for each ordered pair (A,B) of classes considered, we either prove that A@?B holds for every graph or provide a counterexample graph for which A@?B. The seven notions of cycle bases are distinct (either A@?B or B@?A is exhibited for each pair (A,B)). All counterexamples proposed have been designed to be ultimately effective in separating the various algorithmic variants of the MCB problem naturally associated to each one of these seven classes. Furthermore, we provide a linear time algorithm for computing a minimum 2-basis of a graph. Finally, notice that the resolution of the complexity status of some of the remaining three classes would have an immediate impact on practical applications, as for instance in periodic railway timetabling, only integral cycle bases are of direct use.