A Structural Linear-Time Algorithm for Computing the Tutte Decomposition

The block-cut tree decomposes a connected graph along its cutvertices, displaying its 2-connected components. The Tutte-decomposition extends this idea to 2-separators in 2-connected graphs, yielding a canonical tree-decomposition that decomposes the graph into its triconnected components. In 1973, Hopcroft and Tarjan introduced a linear-time algorithm to compute the Tutte-decomposition. Cunningham and Edmonds later established a structural characterization of the Tutte-decomposition via totally-nested 2-separations. We present a conceptually simple algorithm based on this characterization, which computes the Tutte-decomposition in linear time. Our algorithm first computes all totally-nested 2-separations and then builds the Tutte-decomposition from them. Along the way, we derive new structural results on the structure of totally-nested 2-separations in 2-connected graphs using a novel notion of stability, which may be of independent interest.