Abstract
The determination of the isomorphisms between two directed graphs based on those between the corresponding one-input Moore machines plus an additional condition to be checked is developed. Without this condition, the Moore machine isomorphism problem is not equivalent to the graph isomorphism problem. Two algorithms are devised for solving the graph isomorphism problem based on the above method. These algorithms are implemented aspl/1 programs, executed on an IBM System 370/158, experimentally analyzed by a random graph generating program and abal assembly language time routine, and illustrated by examples. The experimental data for the class of isomorphicn/2 regular graphs are fitted by quadratic equations of the input size. In addition the worst-case complexities of both algorithms are found.
Similar content being viewed by others
References
A. T. Berztiss, “A backtrack procedure for isomorphism of directed graphs,”J. ACM 20:365–377 (1973).
D. G. Corneil and C. C. Gotlieb, “An efficient algorithm for graph isomorphism,”J. ACM 17:51–64 (1970).
N. Deo, J. M. Davis, and R. E. Lord, “A new algorithm for digraph isomorphism,”BIT 17:16–30 (1977).
F. Harary, “The determination of the adjacency matrix of a graph,”SIAM Rev. 4:202–210 (1962).
J. E. Hopcraft and J. K. Wong, “Linear time algorithm for isomorphism of planar graphs,”Proceedings of the 6th Annual ACM Symposium on Theory of Computing, Seattle, Washington, 1974, pp. 172–184.
R. M. Karp, “Reducibility among combinatorial problems,” inComplexity of Computer Computations, R. E. Miller and J. W. Thatcher, Eds., Plenum Press, New York (1972), pp. 85–103.
G. Levi, “Graph isomorphism: A heuristic edge-partitioning oriented algorithm,”Computing 12:291–313 (1974).
F. R. Moore, “On the bounds for state-set size in the proofs of equivalence between deterministic, non-deterministic, and two-way finite automata,”IEEE Trans. Computers C-20:1211–1214 (1971).
A. Proskurowski, “Search for a unique incidence matrix of a graph,”BIT 14:209–226 (1974).
G. Salton,Automatic Information Organization and Retrieval, McGraw-Hill, New York, (1968).
D. C. Schmidt and L. E. Druffel, “A fast backtracking algorithm to test directed graphs for isomorphism using distance matrices,”J. ACM 23:433–445 (1976).
Y. J. Shah, G. I. Davids, and M. K. McCarthy, “Optimum features and graph isomorphism,”IEEE Trans. Systems, Mono and Cybernetics 4:313–319 (1974).
E. H. Sussenguth, Jr., “Structure Matching in Information Processing,” Ph.D. Thesis, Harvard University, Cambridge, Massachusetts, 1964.
E. H. Sussenguth, Jr., “A graph-theoretic algorithm for matching chemical structures,”J. Chem., 36–43 (December 5, 1965).
J. Turner, “Generalized matrix functions and the graph isomorphism problem,”SIAM J. Appl. Math. 16:520–526 (1968).
J. R. Ullmann, “Pattern Recognition Techniques,” Crane, Russak and Company, Inc., New York (1973).
J. R. Ullmann, “An algorithm for subgraph isomorphism,”J. ACM 21:31–42 (1976).
S. H. Unger, “GIT-A heuristic program for testing pairs of directed line graphs for isomorphism,”Comm. ACM 7:26–34 (1964).
L. Weinberg, “A simple and efficient algorithm for determining isomorphisms of planar triply connected graphs,“IEEE Trans. Circuit Theory 13:142–148 (1966).
C. C. Yang, “Generation of all closed partitions on a state set of a sequential machine,”IEEE Trans. Computers C-23:530–533 (1974).
C. C. Yang, “Structural preserving morphisms of finite automata and an application to graph isomorphism,”IEEE Trans. Computers C-24:1133–1139 (1975).
C. C. Yang and C. P. May, “A correction and some comments concerning graph isomorphism by finite automata,”IEEE Trans. Computers C-27:95–96 (1978).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Yang, CC., May, C.P. Algorithms for finding directed graph isomorphisms by finite automata. International Journal of Computer and Information Sciences 9, 117–140 (1980). https://doi.org/10.1007/BF00982292
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00982292