Abstract
Given a class of graphs \(\mathcal{G}\), a graph G is a probe graph of \(\mathcal{G}\) if its vertices can be partitioned into two sets ℙ, the probes, and ℕ, the nonprobes, where ℕ is an independent set, such that G can be embedded into a graph of \(\mathcal{G}\) by adding edges between certain vertices of ℕ. If the partition of the vertices into probes and nonprobes is part of the input, then we call the graph a partitioned probe graph of \(\mathcal{G}\). In this paper, we provide a recognition algorithm for partitioned probe permutation graphs with time complexity O(n 2) where n is the number of vertices in the input graph. We show that there are at most O(n 4) minimal separators for a probe permutation graph. As a consequence, there exist polynomial-time algorithms solving treewidth and minimum fill-in problems for probe permutation graphs.
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References
Berry, A., Golumbic, M.C., Lipshteyn, M.: Two tricks to triangulate chordal probe graphs in polynomial time. In: Proceedings of the 15th. In: ACM-SIAM Symposium on Discrete Algorithms, pp. 962–969 (2004)
Bodlaender, H., Kloks, T., Kratsch, D.: Treewidth and Pathwidth of permutation graphs. SIAM Journal on Discrete Mathematics 8, 606–616 (1995)
Booth, K.S., Lueker, G.S.: Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms. Journal of Computer and System Sciences 13, 335–379 (1976)
Bouchitte, V., Todinca, I.: Treewidth and minimum fill-in: Grouping the minimal separators. SIAM Journal on Computing 31, 212–232 (2001)
Chang, M.-S., Kloks, T., Kratsch, D., Liu, J., Peng, S.-L.: On the recognition of probe graphs of some self-complementary graph classes. In: Wang, L. (ed.) COCOON 2005. LNCS, vol. 3595, pp. 808–817. Springer, Heidelberg (2005)
Chang, G.J., Kloks, A.J.J., Liu, J., Peng, S.-L.: The PIGs full monty—A floor show of minimal separators. In: Diekert, V., Durand, B. (eds.) STACS 2005. LNCS, vol. 3404, pp. 521–532. Springer, Heidelberg (2005)
Cowan, D.D., James, L.O., Stanton, R.G.: Graph decomposition for undirected graphs. In: 3rd South-Eastern Conference on Combinatorics, Graph Theory, and Computing, Utilitas Math pp. 281–290 (1972)
Gallai, T.: Transitiv orientierbare Graphen. Acta Math. Sci. Hung. 18, 25–66 (1967)
Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York (1980)
Golumbic, M.C., Lipshsteyn, M.: Chordal probe graphs. Discrete Applied Mathematics 143, 221–237 (2004)
Golumbic, M.C., Trenk, A.N.: Tolerance Graphs, Cambridge studies in advanced mathematics. Cambridge University Press, Cambridge (2004)
Habib, M., de Montgolfier, F., Paul, C.: A simple linear-time modular decomposition algorithm for graphs, using order extensions. In: Hagerup, T., Katajainen, J. (eds.) SWAT 2004. LNCS, vol. 3111, pp. 187–198. Springer, Heidelberg (2004)
Hertz, A.: Slim graphs. Graphs and Combinatorics 5, 149–157 (1989)
Johnson, J.L., Spinrad, J.P.: A polynomial time recognition algorithm for probe interval graphs. In: Proceedings of the 12th ACM-SIAM Symposium on Discrete Algorithms 2001, pp. 477–486 (2001)
Kelly, D.: Comparability graphs. In: Rival, I. (ed.) Graphs and Orders, pp. 3–40 D. Reidel Pub. Comp (1985)
Kloks, T., Kratsch, D.: Listing all minimal separators of a graph. SIAM Journal on Computing 27, 605–613 (1998)
McConnell, R.M., Spinrad, J.P.: Modular decomposition and transitive orientation. Discrete Mathematics 201, 189–241 (1999)
McConnell, R.M., Spinrad, J.: Construction of probe interval graphs. In: Proceedings 13th ACM–SIAM Symposium on Discrete Algorithms 2002, pp. 866–875 (2002)
McMorris, F.R., Wang, C., Zhang, P.: On probe interval graphs. Discrete Applied Mathematics 88, 315–324 (1998)
Möhring, R.H.: Algorithmic aspects of comparability graphs and interval graphs. In: Rival, I. (ed.) Graphs and Orders, pp. 41–101 D. Reidel Pub. Comp. (1985)
Möhring, R.H., Radermacher, F.J.: Substitution decomposition for discrete structures and connections with combinatorial optimization. Annals of Discrete Mathematics 19, 257–356 (1984)
Pnueli, A., Lempel, A., Even, S.: Transitive orientation of graphs and identification of permutation graphs. Canad. J. Math. 23, 160–175 (1971)
Zhang, P.: Probe interval graphs and their application to physical mapping of DNA. Manuscript (1994)
Zhang, P., Schon, E.A., Fisher, S.G., Cayanis, E., Weiss, J., Kistler, S., Bourne, P.E.: An algorithm based on graph theory for the assembly of contigs in physical mapping of DNA. CABIOS 10, 309–317 (1994)
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Chandler, D.B., Chang, MS., Kloks, A.J.J., Liu, J., Peng, SL. (2006). On Probe Permutation Graphs. In: Cai, JY., Cooper, S.B., Li, A. (eds) Theory and Applications of Models of Computation. TAMC 2006. Lecture Notes in Computer Science, vol 3959. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11750321_47
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DOI: https://doi.org/10.1007/11750321_47
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