Abstract
We show that three subclasses of bounded treewidth graphs are well-quasi-ordered by refinements of the minor order. Specifically, we prove that graphs with bounded feedback-vertex-set are well-quasi-ordered by the topological-minor order, graphs with bounded vertex-covers are well-quasi-ordered by the subgraph order, and graphs with bounded circumference are well-quasi-ordered by the induced-minor order. Our results give an algorithm for recognizing any graph family in these classes which is closed under the corresponding minor order refinement.
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References
Arnborg, S.: Efficient algorithms for combinatorial problems on graphs with bounded decomposability. A survey. BIT Numerical Mathematics 25(1), 2–23 (1985)
Arnborg, S., Proskurowski, A.: Linear time algorithms for NP-hard problems restricted to partial k-trees. Discrete Applied Mathematics 23, 11–24 (1989)
Bodlaender, H.L., Fellows, M.R., Hallett, M.T.: Beyond NP-completeness for problems of bounded width: Hardness for the W-hierarchy. In: Proceedings of the 26th annual ACM Symposium on Theory of Computing (STOC), pp. 449–458 (1994)
Corneil, D.G., Keil, J.M.: A dynamic programming approach to the dominating set problem on k-trees. SIAM Journal on Algebraic and Discrete Methods 8(4), 535–543 (1987)
Courcelle, B.: The monadic second-order logic of graphs I. Recognizable sets of finite graphs. Information and Computation 85(1), 12–75 (1990)
Ding, G.: Subgraphs and well-quasi-ordering. Journal of Graph Theory 16(5), 489–502 (1992)
Downey, R., Fellows, M.: Parameterized Complexity. Springer, Heidelberg (1999)
Fellows, M.R., Langston, M.A.: Fast self-reduction algorithms for combinatorial problems of VLSI design. In: Proc. of the 3rd Aegean Workshop On Computing (AWOC), pp. 278–287 (1988)
Fellows, M.R., Langston, M.A.: On well-paritial-order theory and its application to combinatorial problems of VLSI design. SIAM Journal on Discrete Mathematics 5 (1992)
Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006)
Kruskal, J.B.: Well-quasi-ordering, the tree theorem, and Vazsonyi’s conjecture. Transactions of the American Mathematical Society 95, 210–225 (1960)
Menger, K.: Zur allgemeinen kurventheorie. Fundamenta Mathematicae 10, 96–115 (1927)
Nash-Williams, C.S.J.A.: On well-quasi-ordering finite trees. Mathematical Proceedings of the Cambridge Philosophical Society 59(4), 833–835 (1963)
Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, Oxford (2006)
Thomas, R.: Well-quasi-ordering infinite graphs with forbidden finite planar minor. Transactions of the American Mathematical Society 312(1), 67–76 (1990)
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Fellows, M.R., Hermelin, D., Rosamond, F.A. (2009). Well-Quasi-Orders in Subclasses of Bounded Treewidth Graphs. In: Chen, J., Fomin, F.V. (eds) Parameterized and Exact Computation. IWPEC 2009. Lecture Notes in Computer Science, vol 5917. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11269-0_12
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DOI: https://doi.org/10.1007/978-3-642-11269-0_12
Publisher Name: Springer, Berlin, Heidelberg
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