Finding a Minimum Circuit in a Graph

TLDR: Results of Bloniarz, Fisher and Meyer are used to obtain an algorithm with $O(n^2 \log n)$ average behavior, and three methods for finding a triangle in a graph are given.

Abstract: Finding minimum circuits in graphs and digraphs is discussed. An almost minimum circuit is a circuit which may have only one edge more than the minimum. To find an almost minimum circuit an $O(n^2 )$ algorithm is presented. A direct algorithm for finding a minimum circuit has an $O(ne)$ behavior. It is refined to yield an $O(n^2 )$ average time algorithm. An alternative method is to reduce the problem of finding a minimum circuit to that of finding a triangle in an auxiliary graph. Three methods for finding a triangle in a graph are given. The first has an $O(e^{3/2})$ worst case bound ($O(n)$ for planar graphs); the second takes $O(n^{5/3})$ time on the average; the third has an $O(n^{\log 7} )$ worst case behavior. For digraphs, results of Bloniarz, Fisher and Meyer are used to obtain an algorithm with $O(n^2 \log n)$ average behavior.