scholarly journals Rooted Cycle Bases

2017 ◽  
Vol 21 (4) ◽  
pp. 663-686
Author(s):  
David Eppstein ◽  
J. Michael McCarthy ◽  
Brian E. Parrish
Keyword(s):  
Cycle Bases

scholarly journals Mathematical models and a constructive heuristic for finding minimum fundamental cycle bases

2005 ◽  
Vol 15 (1) ◽  
pp. 15-24 ◽  
Author(s):  
Leo Liberti ◽  
Edoardo Amaldi ◽  
Francesco Maffioli ◽  
Nelson Maculan
Keyword(s):  
Linear Programming ◽  
Total Cost ◽  
Fundamental Cycle ◽  
Cycle Basis ◽  
Cpu Time ◽  
Constructive Heuristic ◽  
Cycle Bases ◽  
Time Required ◽  
Application Fields

The problem of finding a fundamental cycle basis with minimum total cost in a graph arises in many application fields. In this paper we present some integer linear programming formulations and we compare their performances, in terms of instance size, CPU time required for the solution, and quality of the associated lower bound derived by solving the corresponding continuous relaxations. Since only very small instances can be solved to optimality with these formulations and very large instances occur in a number of applications, we present a new constructive heuristic and compare it with alternative heuristics.

scholarly journals Matchings, Cycle Bases, and the Maximum Genus of a Graph

10.37236/2479 ◽  
2012 ◽  
Vol 19 (3) ◽  
Author(s):  
Michal Kotrbčík ◽  
Martin Škoviera
Keyword(s):  
Spanning Tree ◽  
Spanning Trees ◽  
Perfect Matching ◽  
Intersection Graph ◽  
Matching Number ◽  
Maximum Genus ◽  
Cycle Space ◽  
Intersection Graphs ◽  
Connected Graphs ◽  
Cycle Bases

We study the interplay between the maximum genus of a graph and bases of its cycle space via the corresponding intersection graph. Our main results show that the matching number of the intersection graph is independent of the basis precisely when the graph is upper-embeddable, and completely describe the range of matching numbers when the graph is not upper-embeddable. Particular attention is paid to cycle bases consisting of fundamental cycles with respect to a given spanning tree. For $4$-edge-connected graphs, the intersection graph with respect to any spanning tree (and, in fact, with respect to any basis) has either a perfect matching or a matching missing exactly one vertex. We show that if a graph is not $4$-edge-connected, different spanning trees may lead to intersection graphs with different matching numbers. We also show that there exist $2$-edge connected graphs for which the set of values of matching numbers of their intersection graphs contains arbitrarily large gaps.

Lower bounds for strictly fundamental cycle bases in grid graphs

Networks ◽  
2009 ◽  
Vol 53 (2) ◽  
pp. 191-205 ◽  
Author(s):  
Ekkehard Köhler ◽  
Christian Liebchen ◽  
Gregor Wünsch ◽  
Romeo Rizzi
Keyword(s):  
Lower Bounds ◽  
Grid Graphs ◽  
Fundamental Cycle ◽  
Cycle Bases

scholarly journals Cycle Bases for Lattices of Binary Matroids with No Fano Dual Minor and Their One-Element Extensions

1999 ◽  
Vol 77 (1) ◽  
pp. 25-38
Author(s):  
Tamás Fleiner ◽  
Winfried Hochstättler ◽  
Monique Laurent ◽  
Martin Loebl
Keyword(s):  
Binary Matroids ◽  
Cycle Bases

Minimum Cycle Bases and Their Applications

2009 ◽  
pp. 34-49 ◽  
Author(s):  
Franziska Berger ◽  
Peter Gritzmann ◽  
Sven de Vries
Keyword(s):  
Cycle Bases

On Optimum Cycle Bases

2001 ◽  
Vol 10 ◽  
pp. 113-116
Author(s):  
Giulia Galbiati
Keyword(s):  
Cycle Bases
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