Abstract
Since its introduction to recognize chordal graphs by Rose, Tarjan, and Lueker, Lexicographic Breadth First Search (LexBFS) has been used to come up with simple, often linear time, algorithms on various classes of graphs. These algorithms are usually multi-sweep algorithms; that is they compute LexBFS orderings \(\sigma _1, \ldots , \sigma _k\), where \(\sigma _i\) is used to break ties for \(\sigma _{i+1}\). Since the number of LexBFS orderings for a graph is finite, this infinite sequence \(\{\sigma _i\}\) must have a loop, i.e. a multi-sweep algorithm will loop back to compute \(\sigma _j\), for some j. We study this new graph invariant, LexCycle(G), defined as the maximum length of a cycle of vertex orderings obtained via a sequence of LexBFS\(^+\). In this work, we focus on graph classes with small LexCycle. We give evidence that a small LexCycle often leads to linear structure that has been exploited algorithmically on a number of graph classes. In particular, we show that for proper interval, interval, co-bipartite, domino-free cocomparability graphs, as well as trees, there exists two orderings \(\sigma \) and \(\tau \) such that \(\sigma = \text {LexBFS}^+(\tau )\) and \(\tau = \text {LexBFS}^+(\sigma )\). One of the consequences of these results is the simplest algorithm to compute a transitive orientation for these graph classes. It was conjectured by Stacho [2015] that LexCycle is at most the asteroidal number of the graph class, we disprove this conjecture by giving a construction for which \({{\mathrm{LexCycle}}}(G) > an(G)\), the asteroidal number of G.
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References
Charbit, P., Habib, M., Mouatadid, L., Naserasr, R.: A new graph parameter to measure linearity. arXiv preprint arXiv:1702.02133 (2017)
Corneil, D.G.: A simple 3-sweep LBFS algorithm for the recognition of unit interval graphs. Discrete Appl. Math. 138(3), 371–379 (2004)
Corneil, D.G., Dalton, B., Habib, M.: LDFS-based certifying algorithm for the minimum path cover problem on cocomparability graphs. SIAM J. Comput. 42(3), 792–807 (2013)
Corneil, D.G., Dusart, J., Habib, M., Kohler, E.: On the power of graph searching for cocomparability graphs. SIAM J. Discret. Math. 30(1), 569–591 (2016)
Derek, G., Corneil, D.G., Olariu, S., Stewart, L.: Linear time algorithms for dominating pairs in asteroidal triple-free graphs. SIAM J. Comput. 28(4), 1284–1297 (1999)
Derek, G., Corneil, D.G., Olariu, S., Stewart, L.: The LBFS structure and recognition of interval graphs. SIAM J. Discret. Math. 23(4), 1905–1953 (2009)
Dragan, F.F., Nicolai, F., Brandstädt, A.: LexBFS-orderings and powers of graphs. In: d’Amore, F., Franciosa, P.G., Marchetti-Spaccamela, A. (eds.) WG 1996. LNCS, vol. 1197, pp. 166–180. Springer, Heidelberg (1997). https://doi.org/10.1007/3-540-62559-3_15
Dusart, J., Habib, M.: A new LBFS-based algorithm for cocomparability graph recognition. Discret. Appl. Math. 216, 149–161 (2017)
Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs, vol. 57. Elsevier (2004)
Monma, C.L., Trotter, W.T.: Tolerance graphs. Discrete Appl. Math. 9(2), 157–170 (1984)
Habib, M., McConnell, R., Paul, C., Viennot, L.: Lex-BFS and partition refinement, with applications to transitive orientation, interval graph recognition and consecutive ones testing. Theor. Comput. Sci. 234(1), 59–84 (2000)
Habib, M., Mouatadid, L.: Maximum induced matching algorithms via vertex ordering characterizations. In: ISAAC 2017 (2017, to appear)
Köhler, E., Mouatadid, L.: Linear time lexDFS on cocomparability graphs. In: Ravi, R., Gørtz, I.L. (eds.) SWAT 2014. LNCS, vol. 8503, pp. 319–330. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-08404-6_28
Köhler, E., Mouatadid, L.: A linear time algorithm to compute a maximum weighted independent set on cocomparability graphs. Inf. Process. Lett. 116(6), 391–395 (2016)
Kratsch, D., McConnell, R.M., Mehlhorn, K., Spinrad, J.P.: Certifying algorithms for recognizing interval graphs and permutation graphs. SIAM J. Comput. 36(2), 326–353 (2006)
Lekkeikerker, C., Boland, J.: Representation of a finite graph by a set of intervals on the real line. Fundamenta Mathematicae 51(1), 45–64 (1962)
McConnell, R.M., Spinrad, J.P.: Modular decomposition and transitive orientation. Discret. Math. 201(1–3), 189–241 (1999)
Rose, D.J., Tarjan, R.E., Lueker, G.S.: Algorithmic aspects of vertex elimination on graphs. SIAM J. Comput. 5(2), 266–283 (1976)
Stacho, J:. Private Communication
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Charbit, P., Habib, M., Mouatadid, L., Naserasr, R. (2017). A New Graph Parameter to Measure Linearity. In: Gao, X., Du, H., Han, M. (eds) Combinatorial Optimization and Applications. COCOA 2017. Lecture Notes in Computer Science(), vol 10628. Springer, Cham. https://doi.org/10.1007/978-3-319-71147-8_11
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DOI: https://doi.org/10.1007/978-3-319-71147-8_11
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