Abstract
A graph class is sandwich monotone if, for every pair of its graphs G 1 = (V,E 1) and G 2 = (V,E 2) with E 1 ⊂ E 2, there is an ordering e 1, ..., e k of the edges in E 2 ∖ E 1 such that G = (V, E 1 ∪ {e 1, ..., e i }) belongs to the class for every i between 1 and k. In this paper we show that strongly chordal graphs and chordal bipartite graphs are sandwich monotone, answering an open question by Bakonyi and Bono from 1997. So far, very few classes have been proved to be sandwich monotone, and the most famous of these are chordal graphs. Sandwich monotonicity of a graph class implies that minimal completions of arbitrary graphs into that class can be recognized and computed in polynomial time. For minimal completions into strongly chordal or chordal bipartite graphs no polynomial-time algorithm has been known. With our results such algorithms follow for both classes. In addition, from our results it follows that all strongly chordal graphs and all chordal bipartite graphs with edge constraints can be listed efficiently.
This work is supported by the Research Council of Norway and National Security Agency, USA.
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References
Alon, N., Shapira, A.: Every monotone graph property is testable. In: STOC 2005, pp. 128–137 (2005)
Bakonyi, M., Bono, A.: Several results on chordal bipartite graphs. Czechoslovak Math. J. 46, 577–583 (1997)
Balogh, J., Bolobás, B., Weinreich, D.: Measures on monotone properties of graphs. Disc. Appl. Math. 116, 17–36 (2002)
Burzyn, P., Bonomo, F., Duran, G.: NP-completeness results for edge modification problems. Discrete Applied Math. 99, 367–400 (2000)
Bodlaender, H.L., Koster, A.M.C.A.: Safe separators for treewidth. Discrete Math. 306, 337–350 (2006)
Bouchitté, V., Todinca, I.: Treewidth and minimum fill-in: Grouping the minimal separators. SIAM J. Comput. 31, 212–232 (2001)
Dahlhaus, E.: Chordale graphen im besonderen hinblick auf parallele algorithmen, Habilitation thesis, Universität Bonn (1991)
Farber, M.: Characterizations on strongly chordal graphs. Discrete Mathematics 43, 173–189 (1983)
de Figueiredo, C.M.H., Faria, L., Klein, S., Sritharan, R.: On the complexity of the sandwich problems for strongly chordal graphs and chordal bipartite graphs. Theoretical Computer Science 381, 57–67 (2007)
Fomin, F.V., Kratsch, D., Todinca, I., Villanger, Y.: Exact algorithms for treewidth and minimum fill-in. SIAM J. Computing 38, 1058–1079 (2008)
Goldberg, P.W., Golumbic, M.C., Kaplan, H., Shamir, R.: Four strikes against physical mapping of DNA. J. Comput. Bio. 2(1), 139–152 (1995)
Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs, 2nd edn. Annals of Discrete Mathematics, vol. 57. Elsevier, Amsterdam (2004)
Heggernes, P., Mancini, F.: Minimal split completions. Discrete Applied Mathematics (in print); also In: Correa, J.R., Hevia, A., Kiwi, M. (eds.) LATIN 2006. LNCS, vol. 3887, pp. 592–604. Springer, Heidelberg (2006)
Heggernes, P., Suchan, K., Todinca, I., Villanger, Y.: Characterizing minimal interval completions: Towards better understanding of profile and pathwidth. In: Thomas, W., Weil, P. (eds.) STACS 2007. LNCS, vol. 4393, pp. 236–247. Springer, Heidelberg (2007)
Heggernes, P., Papadopoulos, C.: Single-Edge Monotonic Sequences of Graphs and Linear-Time Algorithms for Minimal Completions and Deletions. Theoretical Computer Science 410, 1–15 (2009); also In: Lin, G. (ed.) COCOON 2007. LNCS, vol. 4598, pp. 406–416. Springer, Heidelberg (2007)
Kaplan, H., Shamir, R., Tarjan, R.E.: Tractability of parameterized completion problems on chordal, strongly chordal, and proper interval graphs. SIAM J. Comput. 28, 1906–1922 (1999)
Kijima, S., Kiyomi, M., Okamoto, Y., Uno, T.: On listing, sampling, and counting the chordal graphs with edge constraints. In: Hu, X., Wang, J. (eds.) COCOON 2008. LNCS, vol. 5092, pp. 458–467. Springer, Heidelberg (2008)
Lokshtanov, D., Mancini, F., Papadopoulos, C.: Characterizing and Computing Minimal Cograph Completions. In: Preparata, F.P., Wu, X., Yin, J. (eds.) FAW 2008. LNCS, vol. 5059, pp. 147–158. Springer, Heidelberg (2008)
Natanzon, A., Shamir, R., Sharan, R.: Complexity classification of some edge modification problems. Disc. Appl. Math. 113, 109–128 (2001)
Paige, R., Tarjan, R.E.: Three partition refinement algorithms. SIAM J. Comput. 16, 973–989 (1987)
Rose, D.: Triangulated graphs and the elimination process. J. Math. Anal. Appl. 32, 597–609 (1970)
Rose, D.J.: A graph-theoretic study of the numerical solution of sparse positive definite systems of linear equations. In: Read, R.C. (ed.) Graph Theory and Computing, pp. 183–217. Academic Press, New York (1972)
Rose, D., Tarjan, R.E., Lueker, G.: Algorithmic aspects of vertex elimination on graphs. SIAM J. Comput. 5, 266–283 (1976)
Sritharan, R.: Chordal bipartite completion of colored graphs. Discrete Mathematics 308, 2581–2588 (2008)
Spinrad, J.P.: Doubly lexical ordering of dense 0-1 matrices. Information Processing Letters 45, 229–235 (1993)
Yannakakis, M.: Computing the minimum fill-in is NP-complete. SIAM J. Alg. Disc. Meth. 2, 77–79 (1981)
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Heggernes, P., Mancini, F., Papadopoulos, C., Sritharan, R. (2009). Strongly Chordal and Chordal Bipartite Graphs Are Sandwich Monotone. In: Ngo, H.Q. (eds) Computing and Combinatorics. COCOON 2009. Lecture Notes in Computer Science, vol 5609. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02882-3_40
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DOI: https://doi.org/10.1007/978-3-642-02882-3_40
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