Covering a graph by circuits
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(n2), 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.
KeywordsSpan Tree Input Graph Total Execution Time Probabilistic Algorithm Euler Graph
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