Abstract
The Steiner path problem is a restriction of the well known Steiner tree problem such that the required terminal vertices lie on a path of minimum cost. While a Steiner tree always exists within connected graphs, it is not always possible to find a Steiner path. Despite this, one can ask for the Steiner path cover, i.e. a set of vertex disjoint simple paths which contains all terminal vertices and possibly some of the non-terminal vertices. We show how a Steiner path cover of minimum cardinality for the disjoint union and join composition of two graphs can be computed in linear time from the corresponding values of the involved graphs. The cost of an optimal Steiner path cover is the minimum number of Steiner vertices in a Steiner path cover of minimum cardinality. We compute recursively in linear time the cost within a Steiner path cover for the disjoint union and join composition of two graphs by the costs of the involved graphs. This leads us to a linear time computation of an optimal Steiner path, if it exists, for special co-graphs.
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References
Abu-Affash, A.K., Carmi, P., Katz, M.J., Segal, M.: The Euclidean bottleneck Steiner path problem and other applications of (α,β)-pair decomposition. Discrete Comput. Geom. 51(1), 1–23 (2014)
Bodlaender, H., Cygan, M., Kratsch, S., Nederlof, J.: Deterministic single exponential time algorithms for connectivity problems parameterized by treewidth. Inf. Comput. 243, 86–111 (2015)
Corneil, D., Lerchs, H., Stewart-Burlingham, L.: Complement reducible graphs. Discrete Appl. Math. 3, 163–174 (1981)
Corneil, D., Perl, Y., Stewart, L.: A linear recognition algorithm for cographs. SIAM J. Comput. 14(4), 926–934 (1985)
Crespelle, C., Paul, C.: Fully dynamic recognition algorithm and certificate for directed cographs. Discrete Appl. Math. 154(12), 1722–1741 (2006)
Custic, A., Lendl, S.: On streaming algorithms for the Steiner cycle and path cover problem on interval graphs and falling platforms in video games. ACM Comput. Res. Repository abs/1802.08577, 9 pp. (2018)
Moharana, S.S., Joshi, A., Vijay, S.: Steiner path for trees. Int. J. Comput. Appl. 76(5), 11–14 (2013)
Reich, G., Widmayer, P.: Beyond Steiner’s problem: a VLSI oriented generalization. In: Proceedings of Graph-Theoretical Concepts in Computer Science (WG). Lecture Notes in Computer Science, vol. 411, pp. 196–210. Springer, Berlin (1990)
Wald, J., Colbourn, C.: Steiner trees in outerplanar graphs. In: Thirteenth Southeastern Conference on Combinatorics, Graph Theory, and Computing, pp. 15–22 (1982)
Wald, J., Colbourn, C.: Steiner trees, partial 2-trees, and minimum IFI networks. Networks 13, 159–167 (1983)
Westbrook, J., Yan, D.: Approximation algorithms for the class Steiner tree problem (1995). Research Report
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Gurski, F., Hoffmann, S., Komander, D., Rehs, C., Rethmann, J., Wanke, E. (2020). Exact Solutions for the Steiner Path Cover Problem on Special Graph Classes. In: Neufeld, J.S., Buscher, U., Lasch, R., Möst, D., Schönberger, J. (eds) Operations Research Proceedings 2019. Operations Research Proceedings. Springer, Cham. https://doi.org/10.1007/978-3-030-48439-2_40
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DOI: https://doi.org/10.1007/978-3-030-48439-2_40
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