On computing tree and path decompositions with metric constraints on the bags

We here investigate on the complexity of computing the tree-length and the tree-breadth of any graph G, that are respectively the best possible upper-bounds on the diameter and the radius of the bags in a tree decomposition of G. Path-length and path-breadth are similarly defined and studied for path decompositions. So far, it was already known that tree-length is NP-hard to compute. We here prove it is also the case for tree-breadth, path-length and path-breadth. Furthermore, we provide a more detailed analysis on the complexity of computing the tree-breadth. In particular, we show that graphs with tree-breadth one are in some sense the hardest instances for the problem of computing the tree-breadth. We give new properties of graphs with tree-breadth one. Then we use these properties in order to recognize in polynomial-time all graphs with tree-breadth one that are planar or bipartite graphs. On the way, we relate tree-breadth with the notion of k-good tree decompositions (for k = 1), that have been introduced in former work for routing. As a byproduct of the above relation, we prove that deciding on the existence of a k-good tree decomposition is NP-complete (even if k = 1). All this answers open questions from the literature.