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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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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