On the Complexity of Computing Treebreadth
- Cite this paper as:
- Ducoffe G., Legay S., Nisse N. (2016) On the Complexity of Computing Treebreadth. In: Mäkinen V., Puglisi S., Salmela L. (eds) Combinatorial Algorithms. IWOCA 2016. Lecture Notes in Computer Science, vol 9843. Springer, Cham
During the last decade, metric properties of the bags of tree-decompositions of graphs have been studied. Roughly, the length and the breadth of a tree-decomposition are the maximum diameter and radius of its bags respectively. The treelength and the treebreadth of a graph are the minimum length and breadth of its tree-decompositions respectively. Pathlength and pathbreadth are defined similarly for path-decompositions. In this paper, we answer open questions of [Dragan and Köhler, Algorithmica 2014] and [Dragan, Köhler and Leitert, SWAT 2014] about the computational complexity of treebreadth, pathbreadth and pathlength. Namely, we prove that computing these graph invariants is NP-hard. We further investigate graphs with treebreadth one, i.e., graphs that admit a tree-decomposition where each bag has a dominating vertex. We show that it is NP-complete to decide whether a graph belongs to this class. We then prove some structural properties of such graphs which allows us to design polynomial-time algorithms to decide whether a bipartite graph, resp., a planar graph, has treebreadth one.