The cycle space of a strongly connected graph has a basis consisting of directed circuits. The concept of relevant circuits is introduced as a generalization of the relevant cycles in undirected graphs. A polynomial time algorithm for the computation of a minimum weight directed circuit basis is outlined.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1200,
author = {Petra M. Gleiss and Josef Leydold and Peter F. Stadler},
title = {Circuit bases of strongly connected digraphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {23},
year = {2003},
pages = {241-260},
zbl = {1055.05068},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1200}
}
Petra M. Gleiss; Josef Leydold; Peter F. Stadler. Circuit bases of strongly connected digraphs. Discussiones Mathematicae Graph Theory, Tome 23 (2003) pp. 241-260. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1200/
[000] [1] A.T. Balaban, D. Farcasiu, and R. B anica, Graphs of multiple 1,2-shifts in carbonium ions and related systems, Rev. Roum. Chem. 11 (1966) 1205-1227.
[001] [2] R. Balducci and R.S. Pearlman, Efficient exact solution of the ring perception problem, J. Chem. Inf. Comput. Sci. 34 (1994) 822-831, doi: 10.1021/ci00020a016.
[002] [3] C. Berge, Graphs (North-Holland, Amsterdam, NL, 1985).
[003] [4] B. Bollobás, Modern Graph Theory (Springer, New York, 1998).
[004] [5] L.O. Chua and L. Chen, On optimally sparse cycle and coboundary basis for a linear graph, IEEE Trans. Circuit Theory 20 (1973) 54-76.
[005] [6] G.M. Downs, V.J. Gillet, J.D. Holliday and M.F. Lynch, Review of ring perception algorithms for chemical graphs, J. Chem. Inf. Comput. Sci. 29 (1989) 172-187, doi: 10.1021/ci00063a007.
[006] [7] D.A. Fell, Understanding the Control of Metabolism (Portland Press, London, 1997).
[007] [8] A. Galluccio and M. Loebl, (p,q)-odd graphs, J. Graph Theory 23 (1996) 175-184, doi: 10.1002/(SICI)1097-0118(199610)23:2<175::AID-JGT8>3.0.CO;2-Q | Zbl 0859.05047
[008] [9] P.M. Gleiss, P.F. Stadler, A. Wagner and D.A. Fell, Relevant cycles in chemical reaction network, Adv. Cmplx. Syst. 4 (2001) 207-226, doi: 10.1142/S0219525901000140. | Zbl 1028.05108
[009] [10] M. Hartmann, H. Schneider and M.H. Schneider, Integral bases and p-twisted digraphs, Europ. J. Combinatorics 16 (1995) 357-369, doi: 10.1016/0195-6698(95)90017-9. | Zbl 0830.05034
[010] [11] D. Hartvigsen, Minimum path bases, J. Algorithms 15 (1993) 125-142, doi: 10.1006/jagm.1993.1033. | Zbl 0784.68041
[011] [12] D. Hartvigsen and R. Mardon, When do short cycles generate the cycle space, J. Combin. Theory (B) 57 (1993) 88-99, doi: 10.1006/jctb.1993.1008. | Zbl 0723.05079
[012] [13] D. Hartvigsen and R. Mardon, The all-pair min-cut problem and the minimum cycle basis problem on planar graphs, SIAM J. Discrete Math. 7 (1994) 403-418, doi: 10.1137/S0895480190177042. | Zbl 0823.90038
[013] [14] J.D. Horton, A polynomial-time algorithm to find the shortest cycle basis of a graph, SIAM J. Comput. 16 (1987) 359-366, doi: 10.1137/0216026. | Zbl 0632.68064
[014] [15] D.B. Johnson, Finding all the elementary circuits of a directed graph, SIAM J. Comput. 4 (1975) 77-84, doi: 10.1137/0204007. | Zbl 0275.05112
[015] [16] A. Kaveh, Structural Mechanics: Graph and Matrix Methods (Research Studies Press, Exeter, UK, 1992). | Zbl 0858.73002
[016] [17] J.B. Kruskal, On the shortest spanning subgraph of a graph and the travelling salesman problem, Proc. Amer. Math. Soc. 7 (1956) 48-49, doi: 10.1090/S0002-9939-1956-0078686-7.
[017] [18] M. Las Vergnas, Sur le nombre de circuits dans un tournnoi fortement connexe, Cahiers C.E.R.O. 17 (1975) 261-265.
[018] [19] D. Marcu, On finding the elementary paths and circuits of a digraph, Polytech. Univ. Bucharest Sci. Bull. Ser. D: Mech. Engrg. 55 (1993) 29-33.
[019] [20] R.T. Rockafellar, Convex Analysis (Princeton Univ. Press, Princeton NJ, 1970). | Zbl 0193.18401
[020] [21] G.F. Stepanec, Basis systems of vector cycles with extremal properties in graphs, Uspekhi Mat. Nauk. 2, 19 (1964) 171-175 (Russian). | Zbl 0151.33402
[021] [22] K. Thulasiraman and M.N.S. Swamy, Graphs: Theory and Algorithms (J. Wiley & Sons, New York, 1992), doi: 10.1002/9781118033104. | Zbl 0766.05001
[022] [23] P. Vismara, Reconnaissance et représentation d'éléments structuraux pour la description d'objets complexes, Application à l'élaboration de stratégies de synthèse en chimie organique (PhD thesis, Université Montpellier II, France, 1995) 95-MON-2-253.
[023] [24] P. Vismara, Union of all the minimum cycle bases of a graph, Electr. J. Combin. 4 (1997) 73-87. Paper No. #R9 (15 pages). | Zbl 0885.05101