Given a (connected) undirected graph G, a set X ⊆ V(G) and integers k and p, the Steiner Subgraph Extension problem asks whether there exists a set S ⊇ X of at most k vertices such that G[S] is a p-edge-connected subgraph. This problem is a natural generalization of the well-studied Steiner Tree problem (set p = 1 and X to be the terminals). In this paper, we initiate the study of Steiner Subgraph Extension from the perspective of parameterized complexity and give a fixed-parameter algorithm (i.e., FPT algorithm) parameterized by k and p on graphs of bounded degeneracy (removing the assumption of bounded degeneracy results in W-hardness). Besides being an independent advance on the parameterized complexity of network design problems, our result has natural applications. In particular, we use our result to obtain new single-exponential FPT algorithms for several vertex-deletion problems studied in the literature, where the goal is to delete a smallest set of vertices such that: (i) the resulting graph belongs to a specified hereditary graph class, and (ii) the deleted set of vertices induces a p-edge-connected subgraph of the input graph.
@InProceedings{eiben_et_al:LIPIcs.MFCS.2023.45, author = {Eiben, Eduard and Majumdar, Diptapriyo and Ramanujan, M. S.}, title = {{Finding a Highly Connected Steiner Subgraph and its Applications}}, booktitle = {48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)}, pages = {45:1--45:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-292-1}, ISSN = {1868-8969}, year = {2023}, volume = {272}, editor = {Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.45}, URN = {urn:nbn:de:0030-drops-185793}, doi = {10.4230/LIPIcs.MFCS.2023.45}, annote = {Keywords: Parameterized Complexity, Steiner Subgraph Extension, p-edge-connected graphs, Matroids, Representative Families} }
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