Enumerating Cycles and (s, t)-Paths in Undirected Graphs

Abstract

We present the first optimal solution to list all the simple cycles in an undirected graph \(G\) with \(n\) vertices and \(m\) edges. Specifically, let \(\fancyscript{C}(G)\) denote the set of all these cycles. For a cycle \(c \in \fancyscript{C}(G)\), let \(|c|\) denote the number of edges in \(c\). Our algorithm requires \(O(m + \sum _{c \in \fancyscript{C}(G)}{|c|})\) time and is asymptotically optimal: \(\varOmega (m)\) time is necessarily required to read \(G\) as input, and \(\varOmega (\sum _{c \in \fancyscript{C}(G)}{|c|})\) time is required to list the output. We also present the first optimal solution to list all the simple paths from \(s\) to \(t\) (shortly, \((s, t)\)-paths) in an undirected graph \(G\). Let \(\fancyscript{P}_{st}(G)\) denote the set of \((s, t)\)-paths in \(G\) and, for an \((s, t)\)-path \(\pi \in \fancyscript{P}_{st}(G)\), let \(|\pi |\) be the number of edges in \(\pi \). Our algorithm lists all the \((s, t)\)-paths in \(G\) optimally in \(O(m + \sum _{\pi \in \fancyscript{P}_{st}(G)}{|\pi |})\) time.