 Dunwoody found a graph-theoretic proof of Stallings' theorem about ends of groups in 1982, by constructing certain tree-like automorphism invariant graph decompositions.This work has been developed to an important theory in the book "Groups acting on graphs", Cambridge University Press, 1989, with Warren Dicks.In 2002 Dunwoody put forward a proposed proof of the Poincaré conjecture.