For a family of graphs F, given a graph G and an integer k, the F-Deletion problem asks whether we can delete at most k vertices from G to obtain a graph in the family F. The F-Deletion problems for all non-trivial families F that satisfy the hereditary property on induced subgraphs are known to be NP-hard by a result of Yannakakis (STOC'78). Ptolemaic graphs are the graphs that satisfy the Ptolemy inequality, and they are the intersection of chordal graphs and distance-hereditary graphs. Equivalently, they form the set of graphs that do not contain any chordless cycles or a gem as an induced subgraph. (A gem is the graph on 5 vertices, where four vertices form an induced path, and the fifth vertex is adjacent to all the vertices of this induced path.) The Ptolemaic Deletion problem is the F-Deletion problem, where F is the family of Ptolemaic graphs. In this paper we study Ptolemaic Deletion from the viewpoint of Kernelization Complexity, and obtain a kernel with 𝒪(k⁶) vertices for the problem.
@InProceedings{agrawal_et_al:LIPIcs.IPEC.2021.1, author = {Agrawal, Akanksha and Anand, Aditya and Saurabh, Saket}, title = {{A Polynomial Kernel for Deletion to Ptolemaic Graphs}}, booktitle = {16th International Symposium on Parameterized and Exact Computation (IPEC 2021)}, pages = {1:1--1:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-216-7}, ISSN = {1868-8969}, year = {2021}, volume = {214}, editor = {Golovach, Petr A. and Zehavi, Meirav}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2021.1}, URN = {urn:nbn:de:0030-drops-153840}, doi = {10.4230/LIPIcs.IPEC.2021.1}, annote = {Keywords: Ptolemaic Deletion, Kernelization, Parameterized Complexity, Gem-free chordal graphs} }
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