On Algorithms for Enumerating All Circuits of a Graph

Authors: Prabhaker Mateti and Narsingh DeoAuthors Info & Affiliations

https://doi.org/10.1137/0205007

Tools

Abstract

A brief description and comparison of all known algorithms for enumerating all circuits of a graph is provided, and upper bounds on computation time of many algorithms are derived. The vector space method of circuit enumeration is discussed. It is proved that

K_{3}, K_{4}, K_{4} - x

and

K_{3, 3}

are the only undirected and reduced graphs which do not have any edge-disjoint unions of circuits.

Keywords

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References

1.

M. T. Ardon, N. R. Malik, A recursive algorithm for generating circuits and related subgraphs, 5th Asilomar Conf, on Circuits and Systems 5, Pacific Grove, Calif., 1971, 279–284, Nov.

2.

A. T. Berztiss, Data structures. Theory and practice, Academic Press, New York, 1971xiii+442

3.

A. T. Berztiss, A K -tree algorithm for simple cycles of a directed graph, Tech. Rep., 73-6, Dept. of Com-puter Sci., Univ. of Pittsburgh, 1973

4.

Dorwin Cartwright, Terry C. Gleason, The number of paths and cycles in a digraph, Psychometrika, 31 (1966), 179–199

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S. G. Chan, W. T. Chang, On the determination of dicircuits and dipaths, Proc. 2nd Hawaii Internat. Conf. System Sci., Honolulu, Hawaii, 1969, 155–158

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J. P. Char, Master circuit matrix, Proc. IEE (London), 115 (1968), 762–770

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J. P. Char, Master circuit matrix, Proc. IEE (London), 117 (1970), 1655–1656

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A. Ehrenfeucht, L. D. Fosdick, L. J. Osterweil, An algorithm for finding the elementary circuits of a directed graph, Tech. Rep., CU-CS-024-73, Dept, of Computer Sci., Univ. of Colorado, Boulder, 1973

12.

R. W. Floyd, Nondeterministic algorithms, J. Assoc. Comput. Mach., 14 (1967), 636–644

13.

Norman E. Gibbs, A cycle generation algorithm for finite undirected linear graphs, J. Assoc. Comput. Mach., 16 (1969), 564–568

14.

J. Hopcroft, R. Tarjan, Efficient algorithms for graph manipulation, Comm. ACM, 16 (1973), 372–378

15.

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Donald B. Johnson, Finding all the elementary circuits of a directed graph, SIAM J. Comput., 4 (1975), 77–84

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Takahiko Kamae, A systematic method of finding all directed circuits and enumerating all directed paths, IEEE Trans. Circuit Theory, CT-14 (1967), 166–172

18.

J. W. Lapatra, B. R. Myers, Algorithms for circuit enumeration, IEEE Internat. Convention Record, 1964 (1964), 368–373

19.

Prabhaker Mateti, Narsingh Deo, On algorithms for finding all circuits of a graph, Department of Computer Science, University of Illinois, Urbana, Ill., 1973iii+41, UIUCDCD-R-73-585 (revised)

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L. M. Maxwell, G. B. Reed, Subgraph identification—Segs, circuits and paths, 8th Midwest Symp. on Circuit Theory, Colorado State Univ., Fort Collins, Colo., 1965, 13-0–13-10, June

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Information & Authors

Information

Published In

cover image SIAM Journal on Computing

SIAM Journal on Computing

Volume 5 • Issue 1 • March 1976

Pages: 90 - 99

DOI: 10.1137/0205007

ISSN (online): 1095-7111

Copyright

Copyright © 1976 © Society for Industrial and Applied Mathematics.

History

Submitted: 13 August 1973

Accepted: 2 April 1975

Published online: 17 February 2012

Keywords

Authors

Affiliations

Prabhaker Mateti

Narsingh Deo

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References

References

1.

M. T. Ardon, N. R. Malik, A recursive algorithm for generating circuits and related subgraphs, 5th Asilomar Conf, on Circuits and Systems 5, Pacific Grove, Calif., 1971, 279–284, Nov.

2.

A. T. Berztiss, Data structures. Theory and practice, Academic Press, New York, 1971xiii+442

3.

A. T. Berztiss, A K -tree algorithm for simple cycles of a directed graph, Tech. Rep., 73-6, Dept. of Com-puter Sci., Univ. of Pittsburgh, 1973

4.

Dorwin Cartwright, Terry C. Gleason, The number of paths and cycles in a digraph, Psychometrika, 31 (1966), 179–199

5.

S. G. Chan, W. T. Chang, On the determination of dicircuits and dipaths, Proc. 2nd Hawaii Internat. Conf. System Sci., Honolulu, Hawaii, 1969, 155–158

6.

J. P. Char, Master circuit matrix, Proc. IEE (London), 115 (1968), 762–770

7.

J. P. Char, Master circuit matrix, Proc. IEE (London), 117 (1970), 1655–1656

8.

S. M. Chase, Analysis of algorithms for finding all spanning trees of a graph, Rep., 401, Dept. of Computer Sci., Univ. of Illinois, Urbana, Ill., 1970, Ph.D. thesis

9.

G. H. Danielson, On finding the simple paths and circuits in a graph, IEEE Trans. Circuit Theory, CT-15 (1968), 294–295

10.

Narsingh Deo, Graph theory with applications to engineering and computer science, Prentice-Hall Inc., Englewood Cliffs, N.J., 1974xvii+478

11.

A. Ehrenfeucht, L. D. Fosdick, L. J. Osterweil, An algorithm for finding the elementary circuits of a directed graph, Tech. Rep., CU-CS-024-73, Dept, of Computer Sci., Univ. of Colorado, Boulder, 1973

12.

R. W. Floyd, Nondeterministic algorithms, J. Assoc. Comput. Mach., 14 (1967), 636–644

13.

Norman E. Gibbs, A cycle generation algorithm for finite undirected linear graphs, J. Assoc. Comput. Mach., 16 (1969), 564–568

14.

J. Hopcroft, R. Tarjan, Efficient algorithms for graph manipulation, Comm. ACM, 16 (1973), 372–378

15.

H. T. Hsu, P. A. Honkanen, A fast minimal storage algorithm for determining all the elementary cycles of a graph, Computer Sci. Dept., Pennsylvania State Univ., University Park, 1972

16.

Donald B. Johnson, Finding all the elementary circuits of a directed graph, SIAM J. Comput., 4 (1975), 77–84

17.

Takahiko Kamae, A systematic method of finding all directed circuits and enumerating all directed paths, IEEE Trans. Circuit Theory, CT-14 (1967), 166–172

18.

J. W. Lapatra, B. R. Myers, Algorithms for circuit enumeration, IEEE Internat. Convention Record, 1964 (1964), 368–373

19.

Prabhaker Mateti, Narsingh Deo, On algorithms for finding all circuits of a graph, Department of Computer Science, University of Illinois, Urbana, Ill., 1973iii+41, UIUCDCD-R-73-585 (revised)

20.

L. M. Maxwell, G. B. Reed, Subgraph identification—Segs, circuits and paths, 8th Midwest Symp. on Circuit Theory, Colorado State Univ., Fort Collins, Colo., 1965, 13-0–13-10, June

21.

R. L. Norman, A matrix method for location of cycles of a directed graph, Amer. Inst. Chem. Engrs. J., 11 (1965), 450–452

22.

K. Paton, An algorithm for finding a fundamental set of cycles for an undirected linear graph, Comm. ACM, 12 (1969), 514–518

23.

Keith Paton, An algorithm for the blocks and cutnodes of a graph, Comm. ACM, 14 (1971), 468–475

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