Counting closed trails

A closed trail is a connected graph whose every vertex is incident to an even number of edges. We give a deterministic algorithm that in time $2^{m/2}\mathrm{poly}(m,n)$ finds the number of closed trails in a given graph G with n vertices and m edges. Moreover, within the same time bound we can determine every possible vertex set of a closed trail in G, together with the associated number of closed trails. Our algorithm can be used to deterministically find the longest cycle in an n‐vertex claw‐free graph in time $2^{n/2}\mathrm{poly}(m,n)$ via a framework presented by Broersma et al. (in press, http://dx.doi.org/10.1007/s00453-011-9576-4) , thus improving both upon the $O({1.66}^n)$ time randomized algorithm for general graphs (Björklund, 2010, http://dx.doi.org/10.1109/FOCS.2010.24), as well as the $O({1.69}^n)$ time deterministic algorithm for claw‐free graphs by Broersma et al.