Before revision:  Previous algorithms for the uniform sampling task were based on a decomposition of H into a collection of odd cycles and stars, denoted D^*(H)=\{O_{k_1}, \ldots, O_{k_q}, S_{p_1}, \ldots, S_{p_\ell}\}.
After revision:  Previous algorithms for these tasks were based on a decomposition of H into a collection of odd cycles and stars, denoted D^*(H)=\{O_{k_1}, \ldots, O_{k_q}, S_{p_1}, \ldots, S_{p_\ell}\}.