# Covering a graph by circuits

## Abstract

A circuit cover is a set of circuits which cover all the edges of a graph; its length is the sum of the lengths of the circuits. In analyzing irrigation systems it is sometimes necessary to find a short circuit cover. It is shown that every bridge-free connected undirected graph with n vertices and e edges has a circuit cover the length of which is less than or equal to e+2nlogn. A probabilistic algorithm for finding such a cover is presented; its expected running time is 0(n^{2}), independent of the input graph. This constitutes an example of solving a graph-theoretical problem by a probabilistic algorithm — the class of algorithms introduced by Rabin.

If the graph contains two edge-disjoint spanning trees then there exists a circuit cover of length at most e+n-1.

The relationship of circuit covers to the Chinese postman problem is also discussed. It is proven that there exist graphs for which the shortest circuit cover is longer than any optimal postman tour.

## Keywords

Span Tree Input Graph Total Execution Time Probabilistic Algorithm Euler Graph## Preview

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## References

- [AHU]A.V. Aho, J.E. Hopcroft and J.D. Ullman, The Design and Analysis of Computer Algorithms, Addison-Wesley, (1974).Google Scholar
- [C]H. Cross, "Analysis of Flow in Networks of Conduits of Conductors", Bull. No. 286, Univ. of Illinois Engineering Experimental Station, Urbana, Ill. (1936).Google Scholar
- [EJ]J. Edmonds and E.L. Johnson, "Matching, Euler Tours and the Chinese Postman", Math. Programming 5 (1973), 88–124.CrossRefGoogle Scholar
- [EK]J. Edmonds, and R.M. Karp, "Theoretical Improvements in Algorithmic Efficiency for Network Flow Problems", J. ACM 19 (1972) 248–264.CrossRefGoogle Scholar
- [K]T. Kameda, "On Maximally Distant Spanning Trees of a Graph", Computing 17 (1976), 115–119.Google Scholar
- [L]C.L. Liu, Introduction to Combinatorial Mathematics, McGraw-Hill (1968).Google Scholar
- [R]M.O. Rabin, "Probabilistic Algorithms", Proc. Sym. on New Directions and Recent Results in Algorithms and Complexity, Carnegie-Mellon University, Academic Press (April 1976).Google Scholar