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Complexity of Finding Embeddings in a k-Tree
Abstract
A k-tree is a graph that can be reduced to the k-complete graph by a sequence of removals of a degree k vertex with completely connected neighbors. We address the problem of determining whether a graph is a partial graph of a k-tree. This problem is motivated by the existence of polynomial time algorithms for many combinatorial problems on graphs when the graph is constrained to be a partial k-tree for fixed k. These algorithms have practical applications in areas such as reliability, concurrent broadcasting and evaluation of queries in a relational database system. We determine the complexity status of two problems related to finding the smallest number k such that a given graph is a partial k-tree. First, the corresponding decision problem is NP-complete. Second, for a fixed (predetermined) value of k, we present an algorithm with polynomially bounded (but exponential in k) worst case time complexity. Previously, this problem had only been solved for $k = 1,2,3$.
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Cited by
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Robertson, N., & Seymour, P. D. (1986) · Journal of Algorithms · doi:10.1016/0196-6774(86)90023-4 · Cited by 1,542
A Linear-Time Algorithm for Finding Tree-Decompositions of Small Treewidth
Bodlaender, H. L. (1996) · SIAM Journal on Computing · doi:10.1137/s0097539793251219 · Cited by 1,542
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Kaess, M., Johannsson, H., Roberts, R., Ila, V., Leonard, J. J., & Dellaert, F. (2012) · The International Journal of Robotics Research · doi:10.1177/0278364911430419 · Cited by 1,362
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Niedermeier, R. (2006) · Invitation to Fixed-Parameter Algorithms · doi:10.1093/acprof:oso/9780198566076.003.0013 · Cited by 574
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Markov, I. L., & Shi, Y. (2008) · SIAM Journal on Computing · doi:10.1137/050644756 · Cited by 472
Metadata
- Authors
- Arnborg, S., Corneil, D. G., & Proskurowski, A.
- Venue
- SIAM Journal on Algebraic Discrete Methods
- Published
- 1987-04
- Volume
- 8
- Issue
- 2
- Pages
- 277-284
- ISSN
- 0196-5212 (ISSN-L)
- 2168-345X
- OpenAlex
- W2085550081
- Cited by
- 1,206
- References
- 0
- FWCI
- 18.84
- Type
- journal article
- Language
- en
- Open access
- closed
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- CoChat. (2026). Citation check: Complexity of Finding Embeddings in a k-Tree. https://app.cochat.ai/papers/10.1137/0608024
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- Arnborg, S., Corneil, D. G., & Proskurowski, A. (1987). Complexity of Finding Embeddings in a k-Tree. SIAM Journal on Algebraic Discrete Methods. https://doi.org/10.1137/0608024
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