Abstract
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.
This work is partially supported by ANR project Stint under reference ANR-13-BS02-0007 and ANR program “Investments for the Future” under reference ANR-11-LABX-0031-01.
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Notes
- 1.
We aim at turning the separator N(v) into a clique. However, we cannot do that directly since it would break the distances in G, and the graph needs to stay planar.
- 2.
When v is of Type 1 we call the algorithm on \(G'\), obtained from \(G {\setminus } v\) by contracting the internal nodes of \(\varPi _v\) to an edge, in order to obtain a quadratic complexity.
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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. https://doi.org/10.1007/978-3-319-44543-4_1
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