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Difference Determines the Degree: Structural Kernelizations of Component Order Connectivity

Authors Sriram Bhyravarapu, Satyabrata Jana , Saket Saurabh , Roohani Sharma



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Author Details

Sriram Bhyravarapu
  • The Institute of Mathematical Sciences, HBNI, Chennai, India
Satyabrata Jana
  • The Institute of Mathematical Sciences, HBNI, Chennai, India
Saket Saurabh
  • The Institute of Mathematical Sciences, HBNI, Chennai, India
  • University of Bergen, Norway
Roohani Sharma
  • Max Planck Institute for Informatics, Saarland Informatics Campus, Saarbrücken, Germany

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Sriram Bhyravarapu, Satyabrata Jana, Saket Saurabh, and Roohani Sharma. Difference Determines the Degree: Structural Kernelizations of Component Order Connectivity. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 5:1-5:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.IPEC.2023.5

Abstract

We consider the question of polynomial kernelization of a generalization of the classical Vertex Cover problem parameterized by a parameter that is provably smaller than the solution size. In particular, we focus on the c-Component Order Connectivity problem (c-COC) where given an undirected graph G and a non-negative integer t, the objective is to test whether there exists a set S of size at most t such that every component of G-S contains at most c vertices. Such a set S is called a c-coc set. It is known that c-COC admits a kernel with {O}(ct) vertices. Observe that for c = 1, this corresponds to the Vertex Cover problem. We study the c-Component Order Connectivity problem parameterized by the size of a d-coc set (c-COC/d-COC), where c,d ∈ ℕ with c ≤ d. In particular, the input is an undirected graph G, a positive integer t and a set M of at most k vertices of G, such that the size of each connected component in G - M is at most d. The question is to find a set S of vertices of size at most t, such that the size of each connected component in G - S is at most c. In this paper, we give a kernel for c-COC/d-COC with O(k^{d-c+1}) vertices and O(k^{d-c+2}) edges. Our result exhibits that the difference in d and c, and not their absolute values, determines the exact degree of the polynomial in the kernel size. When c = d = 1, the c-COC/d-COC problem is exactly the Vertex Cover problem parameterized by the solution size, which has a kernel with O(k) vertices and O(k²) edges, and this is asymptotically tight [Dell & Melkebeek, JACM 2014]. We also show that the dependence of d-c in the exponent of the kernel size cannot be avoided under reasonable complexity assumptions.

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
Keywords
  • Kernelization
  • Component Order Connectivity
  • Vertex Cover
  • Structural Parameterizations

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