An algorithm for finding a fundamental set of cycles of a graph

ABSTRACT

A fast method is presented for finding a fundamental set of cycles for an undirected finite graph. A spanning tree is grown and the vertices examined in turn, unexamined vertices being stored in a pushdown list to await examination. One stage in the process is to take the top element v of the pushdown list and examine it, i.e. inspect all those edges (v, z) of the graph for which z has not yet been examined. If z is already in the tree, a fundamental cycle is added; if not, the edge (v, z) is placed in the tree. There is exactly one such stage for each of the n vertices of the graph. For large n, the store required increases as n2 and the time as n&ggr; where &ggr; depends on the type of graph involved. &ggr; is bounded below by 2 and above by 3, and it is shown that both bounds are attained. In terms of storage our algorithm is similar to that of Gotlieb and Corneil and superior to that of Welch; in terms of speed it is similar to that of Welch and superior to that of Gotlieb and Corneil. Tests show our algorithm to be remarkably efficient (&ggr; = 2) on random graphs.

REFERENCES

• 1: GOULD, R. The application of graph theory to the synthesis of contact networks. Proc. International Symp. on the Theory of Switching, Pt. I, Apr. 2-5, 1957. In The Annals of the Computation Laboratory of Harvard University, Annals No. 29, Harvard U. Press, Cambridge, Mass., 1959, pp. 244-292.
• 2: John T. Welch, Jr., A Mechanical Analysis of the Cyclic Structure of Undirected Linear Graphs, Journal of the ACM (JACM), v.13 n.2, p.205-210, April 1966 [doi>10.1145/321328.321331]
• 3: C. G. Gotlieb , D. G. Corneil, Algorithms for finding a fundamental set of cycles for an undirected linear graph, Communications of the ACM, v.10 n.12, p.780-783, Dec. 1967 [doi>10.1145/363848.363861]
• 4: SUSSENGUTH, E., JR. A graph theoretical algorithm for matching chemical structures. J. Chem. Doe. 5, 1 (Feb. 1965), 36-43.

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.2 DISCRETE MATHEMATICS

Additional Classification:
  F. Theory of Computation
  F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY
      F.2.2 Nonnumerical Algorithms and Problems
          Subjects: Computations on discrete structures

General Terms:
Algorithms

Keywords:
algorithm, cycle, fundamental cycle set, graph, spanning tree