Fixed-parameter tractability and characterizations of small special treewidth

We investigate fixed-parameter aspects of the notion of special treewidth, which was recently introduced by Courcelle [8,9]. In a special tree decomposition, for each vertex v, the bags containing v form a rooted path in decomposition tree. We resolve an open problem by Courcelle, and show that an algorithm by Bodlaender and Kloks [7] can be modified to obtain for each fixed k, a linear time algorithm that decides if the special treewidth of a given graph is at most k, and if so, finds a corresponding special tree decomposition. This establishes that special treewidth is fixed-parameter tractable. We obtain characterizations for the class of graphs of special treewidth at most two. The first characterization consists of certain linear structures (termed mambas, or equivalently, biconnected partial two-paths) arranged in a specific tree-like fashion, building upon characterizations of biconnected graphs of treewidth two or of pathwidth two. We show that the class of graphs of special treewidth at most two is closed under taking of minors, and give explicitly the obstruction set for this class. For k ≥ 3, the class of graphs of special treewidth at most k is not closed under taking minors.