On computing tree and path decompositions with metric constraints on the bags
Abstract
We here investigate on the complexity of computing the \emph{treelength} and the \emph{treebreadth} of any graph $G$, that are respectively the best possible upperbounds on the diameter and the radius of the bags in a tree decomposition of $G$. \emph{Pathlength} and \emph{pathbreadth} are similarly defined and studied for path decompositions. So far, it was already known that treelength is NPhard to compute. We here prove it is also the case for treebreadth, pathlength and pathbreadth. Furthermore, we provide a more detailed analysis on the complexity of computing the treebreadth. In particular, we show that graphs with treebreadth one are in some sense the hardest instances for the problem of computing the treebreadth. We give new properties of graphs with treebreadth one. Then we use these properties in order to recognize in polynomialtime all graphs with treebreadth one that are planar or bipartite graphs. On the way, we relate treebreadth with the notion of \emph{$k$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 NPcomplete (even if $k=1$). All this answers open questions from the literature.
 Publication:

arXiv eprints
 Pub Date:
 January 2016
 arXiv:
 arXiv:1601.01958
 Bibcode:
 2016arXiv160101958D
 Keywords:

 Computer Science  Computational Complexity;
 Computer Science  Data Structures and Algorithms
 EPrint:
 50 pages, 39 figures