A Polynomial Kernel for Deletion to Ptolemaic Graphs

Authors Akanksha Agrawal , Aditya Anand, Saket Saurabh

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Author Details

Akanksha Agrawal
  • Indian Institute of Technology Madras, Chennai, India
Aditya Anand
  • Indian Institute of Technology Kharagpur, India
Saket Saurabh
  • Institute of Mathematical Sciences, Chennai, India
  • University of Bergen, Norway

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Akanksha Agrawal, Aditya Anand, and Saket Saurabh. A Polynomial Kernel for Deletion to Ptolemaic Graphs. In 16th International Symposium on Parameterized and Exact Computation (IPEC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 214, pp. 1:1-1:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Ptolemaic Deletion
  • Kernelization
  • Parameterized Complexity
  • Gem-free chordal graphs


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