An improved exact algorithm for undirected feedback vertex set

A feedback vertex set in an undirected graph is a subset of vertices removal of which leaves a graph with no cycles. Razgon (in: Proceedings of the 10th Scandinavian workshop on algorithm theory (SWAT 2006), pp. 160–171, 2006) gave a 1.8899nnO(1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1.8899^n n^{O(1)}$$\end{document}-time algorithm for finding a minimum feedback vertex set in an n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document}-vertex undirected graph, which is the first exact algorithm for the problem that breaks the trivial barrier of 2n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2^n$$\end{document}. Later, Fomin et al. (Algorithmica 52:293–307, 2008) improved the result to 1.7548nnO(1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1.7548^n n^{O(1)}$$\end{document}. In this paper, we further improve the result to 1.7266nnO(1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1.7266^n n^{O(1)}$$\end{document}. Our algorithm is analyzed by the measure-and-conquer method. We get the improvement by designing new reductions based on biconnectivity of instances and introducing a new measure scheme on the structure of reduced graphs.