Abstract
Enumerating objects of a specified type is one of the principal tasks in algorithmics. In graph algorithms one often enumerates vertex subsets satisfying a certain property. The optimization problem Tropical Connected Set is strongly related to the Graph Motif problem which deals with vertex-colored graphs and has various applications in metabolic networks and biology. A tropical connected set of a vertex-colored graph is a subset of the vertices which induces a connected subgraph in which all colors of the input graph appear at least once; among others this generalizes steiner trees. We investigate the enumeration of the inclusion-minimal tropical connected sets of a given vertex-colored graph. We present algorithms to enumerate all minimal tropical connected sets on colored graphs of the following graph classes: on split graphs in running in time \(O^*(1.6402^n)\), on interval graphs in \(O^*(1.8613^n)\) time, on cobipartite graphs and block graphs in \(O^*(3^{n/3})\). Our algorithms imply corresponding upper bounds on the number of minimal tropical connected sets in graphs on n vertices for each of these graph classes. We also provide various new lower bound constructions thereby obtaining matching lower bounds for cobipartite and block graphs.
This work is supported by the French National Research Agency (ANR project GraphEn / ANR-15-CE40-0009).
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References
Angles d’Auriac, J.-A., Cohen, N., El Maftouhi, A., Harutyunyan, A., Legay, S., Manoussakis, Y.: Connected tropical subgraphs in vertex-colored graphs. Discrete Math. Theor. Comput. Sci. 17(3), 327–348 (2016). http://dmtcs.episciences.org/2151
Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph classes: a survey. society for industrial and applied mathematics (SIAM), Philadelphia, PA, SIAM Monographs on Discrete Mathematics and Applications (1999)
Chapelle, M., Cochefert, M., Kratsch, D., Letourneur, R., Liedloff, M.: Exact exponential algorithms to find a tropical connected set of minimum size. In: Cygan, M., Heggernes, P. (eds.) IPEC 2014. LNCS, vol. 8894, pp. 147–158. Springer, Heidelberg (2014)
Couturier, J., Heggernes, P., van’t Hof, P., Kratsch, D.: Minimal dominating sets in graph classes, combinatorial bounds and enumeration. Theor. Comput. Sci. 487, 82–94 (2013)
Couturier, J.-F., Heggernes, P., van’t Hof, P., Villanger, Y.: Maximum number of minimal feedback vertex sets in chordal graphs and cographs. In: Gudmundsson, J., Mestre, J., Viglas, T. (eds.) COCOON 2012. LNCS, vol. 7434, pp. 133–144. Springer, Heidelberg (2012)
Couturier, J., Letourneur, R., Liedloff, M.: On the number of minimal dominating sets on some graph classes. Theor. Comput. Sci. 562, 634–642 (2015)
Diestel, R.: Graph Theory. Graduate Texts in Mathematics, vol. 173. Springer, Heidelberg (2012)
Fomin, F.V., Gaspers, S., Pyatkin, A.V., Razgon, I.: On the minimum feedback vertex set problem: exact and enumeration algorithms. Algorithmica 52(2), 293–307 (2008)
Fomin, F.V., Gaspers, S., Lokshtanov, D., Saurabh, S.: Exact algorithms via monotone local search. In: Proceedings of STOC, pp. 764–775 (2016)
Fomin, F.V., Grandoni, F., Pyatkin, A.V., Stepanov, A.A.: Combinatorial bounds via measure, conquer: bounding minimal dominating sets and applications. ACM Trans. Algorithms 5(1), 9 (2008)
Fomin, F.V., Heggernes, P., Kratsch, D.: Exact algorithms for graph homomorphisms. Theor. Comp. Sys. 41(2), 381–393 (2007)
Fomin, F.V., Heggernes, P., Kratsch, D., Papadopoulos, C., Villanger, Y.: Enumerating minimal subset feedback vertex sets. Algorithmica 69(1), 216–231 (2014)
Fomin, F.V., Kratsch, D.: Exact Exponential Algorithms. Texts in Theoretical Computer Science. An EATCS Series. Springer, Heidelberg (2010)
Fomin, F.V., Villanger, Y.: Finding induced subgraphs via minimal triangulations. In: Proceedings STACS 2010, Dagstuhl, LIPIcs, vol. 43, pp. 383–394 (2010)
Gaspers, S., Mackenzie, S.: On the number of minimal separators in graphs, CoRR abs/1503.01203 (2015)
Gaspers, S., Mnich, M.: Feedback vertex sets in tournaments. J. Graph Theor. 72(1), 72–89 (2013)
Golovach, P.A., Heggernes, P., Kratsch, D.: Enumeration and maximum number of minimal connected vertex covers in graphs. In: Lipták, Z., Smyth, W.F. (eds.) IWOCA 2015. LNCS, vol. 9538, pp. 235–247. Springer, Heidelberg (2016)
Golovach, P.A., Heggernes, P., Kratsch, D., Saei, R.: Subset feedback vertex sets in chordal graphs. J. Discr. Algorithms 26, 7–15 (2014)
Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Annals of Discrete Mathematics, vol. 57, 2nd edn. Elsevier Science B.V., Amsterdam (2004)
Kay, D.C., Chartrand, G.: A characterization of certain Ptolemaic graphs. Can. J. Math. 17, 342–346 (1965)
Lacroix, V., Fernandes, C.G., Sagot, M.-F.: Motif search in graphs: application to metabolic networks. IEEE/ACM Trans. Comput. Biol. Bioinform. 3(4), 360–368 (2006)
McMorris, F., Warnow, T., Wimer, T.: Triangulating vertex-colored graphs. SIAM J. Discr. Math. 7, 296–306 (1994)
Moon, J.W., Moser, L.: On cliques in graphs. Israel J. Math. 3, 23–28 (1965)
Scott, J., Ideker, T., Karp, R.M., Sharan, R.: Efficient algorithms for detecting signaling pathways in protein interaction networks. J. Comput. Biol. 13, 133–144 (2006)
Telle, J.A., Villanger, Y.: Connecting terminals and 2-disjoint connected subgraphs. In: Brandstädt, A., Jansen, K., Reischuk, R. (eds.) WG 2013. LNCS, vol. 8165, pp. 418–428. Springer, Heidelberg (2013)
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Kratsch, D., Liedloff, M., Sayadi, M.Y. (2017). Enumerating Minimal Tropical Connected Sets. In: Steffen, B., Baier, C., van den Brand, M., Eder, J., Hinchey, M., Margaria, T. (eds) SOFSEM 2017: Theory and Practice of Computer Science. SOFSEM 2017. Lecture Notes in Computer Science(), vol 10139. Springer, Cham. https://doi.org/10.1007/978-3-319-51963-0_17
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