# Finding Shorter Cycles in a Weighted Graph

## Abstract

In this paper we investigate the structures of short cycles in a weighted graph. By Thomassen’s \(3\)-path-condition theory (Thomassen in J Comb Theory Ser 48:155–177, 1990), it is easy to find a shortest cycle in a collection of cycles beyond a subspace of the cycle space of a graph. What is more difficult for one to do is to find a shortest cycle within a subspace of the cycle space in polynomial time. By using the Dijkstra’s algorithm (Dijkstra in Numer Math 1:55–61, 1959) we find a collection \(\mathcal {C}\) of cycles containing many types of short cycles within a given subspace of the cycle space of a graph and this implies a polynomial time algorithm (called *extended fundamental cycle algorithm*) to locate all the possible shortest cycles in a weighted graph. In the case of unweighted graphs, the algorithm may also find every shortest even cycle in a graph, this greatly improved a result of Grötschel and Pulleyblank (Oper Res Lett 1:23–27, 1981/82), Monien (Computing 31:355–369, 1983), Yuster and Zwick (SIAM J Discrete Math 10:209–222, 1997). In fact, our algorithm shows that there are at most \(O(n^4)\) many such short cycles in an unweighted graph of order \(n\). Further more, our fundamental cycle method may find a minimum cycle base (or simply *MCB* as some scholars named) in the cycle space of a graph. Since the structure of MCB’s is *unique* (Ren and Deng in Discrete Math 307:2654–2660, 2007), this shows that, in the sense, cycles in a MCB are *nearly-fundamental* (i.e., each element in a MCB is a sum of at most two fundamental cycles). This provides a new way to study MCB.

### Keywords

Fundamental cycle Dijkstra’s algorithm \(3\)-Path-condition### References

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