The Plane Diameter Completion problem asks, given a plane graph G and a positive integer d, if it is a spanning subgraph of a plane graph H that has diameter at most d. We examine two variants of this problem where the input comes with another parameter k. In the first variant, called BPDC, k upper bounds the total number of edges to be added and in the second, called BFPDC, k upper bounds the number of additional edges per face. We prove that both problems are NP-complete, the first even for 3-connected graphs of face-degree at most 4 and the second even when k=1 on 3-connected graphs of face-degree at most 5. In this paper we give parameterized algorithms for both problems that run in O(n^{3})+2^{2^{O((kd)^2\log d)}} * n steps.
@InProceedings{golovach_et_al:LIPIcs.IPEC.2015.30, author = {Golovach, Petr A. and Requil\'{e}, Cl\'{e}ment and Thilikos, Dimitrios M.}, title = {{Variants of Plane Diameter Completion}}, booktitle = {10th International Symposium on Parameterized and Exact Computation (IPEC 2015)}, pages = {30--42}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-92-7}, ISSN = {1868-8969}, year = {2015}, volume = {43}, editor = {Husfeldt, Thore and Kanj, Iyad}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2015.30}, URN = {urn:nbn:de:0030-drops-55697}, doi = {10.4230/LIPIcs.IPEC.2015.30}, annote = {Keywords: Planar graphs, graph modification problems, parameterized algorithms, dynamic programming, branchwidth} }
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